#508314

order	512
nilpotency class	4
isoclinism	isoclinic class of #508313
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

672 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 672 conjugacy classes of subgroups — too many to draw. Only its 138 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)(9,10,13,19,32,45,61,58,15,18,34,44,14,20,33,47)(11,16,22,38,23,39,54,69,25,37,56,67,24,40,55,70)(12,26,28,48,35,64,63,42,30,59,62,43,29,46,31,17)(21,49,50,41,27,57,60,65,52,71,72,66,51,68,53,36);(2,5)(6,8)(9,11)(10,16)(12,27)(13,22)(14,24)(15,25)(17,41)(18,37)(19,38)(20,40)(21,29)(23,32)(26,57)(28,60)(30,51)(31,50)(33,55)(34,56)(35,52)(36,43)(39,45)(42,66)(44,67)(46,49)(47,70)(48,65)(53,62)(54,61)(58,69)(59,68)(63,72)(64,71);(1,3)(2,6)(4,7)(5,8)(9,12,15,30)(10,17,18,42)(11,21,25,52)(13,31,34,63)(14,35,32,29)(16,36,37,65)(19,46,44,64)(20,48,45,43)(22,53,56,60)(23,51,24,27)(26,58,59,47)(28,61,62,33)(38,68,67,57)(39,66,40,41)(49,69,71,70)(50,54,72,55);(3,7)(6,8)(10,18)(16,37)(17,42)(19,44)(20,45)(26,59)(36,65)(38,67)(39,40)(41,66)(43,48)(46,64)(47,58)(49,71)(57,68)(69,70)

Polycyclic presentation:

2^9|a^2=f,c^2=i,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,g^b=gi,d^c=de,f^c=fh,g^c=gi,h^c=hi

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, 415510);