#510058

order	512
nilpotency class	4
isoclinism	isoclinic class of #509163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂³	#576
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

466 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 466 conjugacy classes of subgroups — too many to draw. Only its 136 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)(3,4,8,6)(9,10,13,20,14,21,36,41,16,23,38,54,15,22,37,53)(11,17,25,42,39,52,69,19,27,44,60,55,26,43,35,50)(12,28,30,47,31,64,66,49,33,65,67,48,32,51,34,18)(24,56,57,70,61,71,72,63,58,46,68,45,29,62,59,40);(1,3,5,8)(2,6,7,4)(9,11)(10,17)(12,29)(13,25)(14,39)(15,26)(16,27)(18,45)(19,41)(20,42)(21,52)(22,43)(23,44)(24,31)(28,62)(30,59)(32,58)(33,61)(34,68)(35,37)(36,69)(38,60)(40,47)(46,51)(48,63)(49,70)(50,53)(54,55)(56,64)(57,66)(65,71)(67,72);(9,12,16,33)(10,18,23,49)(11,24,27,58)(13,34,38,66)(14,32,15,31)(17,40,44,63)(19,46,50,56)(20,51,54,64)(21,48,22,47)(25,59,60,72)(26,61,39,29)(28,41,65,53)(30,36,67,37)(35,57,69,68)(42,62,55,71)(43,70,52,45);(1,4,5,6)(2,3,7,8)(9,11)(10,19)(12,24)(13,35)(14,26)(15,39)(16,27)(17,41)(18,46)(20,52)(21,42)(22,55)(23,50)(25,37)(28,63)(29,31)(30,59)(32,61)(33,58)(34,57)(36,60)(38,69)(40,65)(43,54)(44,53)(45,51)(47,71)(48,62)(49,56)(64,70)(66,68)(67,72)

Polycyclic presentation:

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

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, 417254);