#508780

order	512
nilpotency class	4
isoclinism	isoclinic class of #508669
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3054	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

546 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 546 conjugacy classes of subgroups — too many to draw. Only its 118 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 72 points:

(1,2)(3,6)(4,5)(7,8)(9,10,14,21,17,24,15,22)(11,25,28,46,29,52,40,18)(12,30,33,19,36,49,34,51)(13,37,38,23,39,20,16,41)(26,44,56,70,57,71,61,55)(27,58,42,45,60,53,65,69)(31,64,59,72,66,47,68,62)(32,50,35,54,43,63,67,48);(2,5)(6,8)(9,11)(10,18)(12,31)(13,27)(14,40)(15,28)(16,42)(17,29)(19,47)(20,45)(21,52)(22,25)(23,53)(24,46)(26,35)(30,62)(32,56)(33,68)(34,59)(36,66)(37,69)(38,65)(39,60)(41,58)(43,61)(44,50)(48,70)(49,72)(51,64)(54,55)(57,67)(63,71);(3,7)(6,8)(9,12,17,36)(10,19,24,51)(11,26,29,57)(13,35,39,67)(14,34,15,33)(16,43,38,32)(18,44,46,71)(20,50,37,63)(21,30,22,49)(23,54,41,48)(25,55,52,70)(27,59,60,68)(28,61,40,56)(31,65,66,42)(45,62,69,72)(47,58,64,53);(1,3,4,7)(2,6,5,8)(9,13,15,16,17,39,14,38)(10,20,22,23,24,37,21,41)(11,27,28,42,29,60,40,65)(12,32,34,67,36,43,33,35)(18,45,25,53,46,69,52,58)(19,48,30,63,51,54,49,50)(26,31,56,59,57,66,61,68)(44,47,70,62,71,64,55,72)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=efhi,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,g^a=gi,c^b=cg,e^b=ei,g^b=gh,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gh,h^c=hi,g^d=gi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 415976);