#499687

order	512
nilpotency class	4
isoclinism	isoclinic class of #499266
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	C₂×AGL(1,ℤ/8)
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

500 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 130 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 406883);