#510055

order	512
nilpotency class	4
isoclinism	isoclinic class of #509155
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

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

normal subgroups only

This group has 468 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)(9,10,18,43,16,24,42,47,14,22,15,23,17,25,41,55)(11,26,37,48,38,49,45,21,13,19,28,59,29,58,40,56)(12,30,64,54,35,67,39,20,33,66,34,53,36,57,44,52)(27,51,70,72,63,65,62,60,32,69,61,50,31,68,71,46);(1,3,5,8)(2,6,7,4)(9,11)(10,19)(12,31)(13,14)(15,28)(16,38)(17,29)(18,37)(20,50)(21,55)(22,26)(23,48)(24,58)(25,49)(27,35)(30,65)(32,36)(33,63)(34,62)(39,70)(40,41)(42,45)(43,59)(44,61)(46,53)(47,56)(51,57)(52,72)(54,60)(64,71)(66,68)(67,69);(9,12,14,33)(10,20,22,52)(11,27,13,32)(15,39,18,44)(16,36,17,35)(19,46,26,60)(21,51,56,69)(23,57,43,67)(24,54,25,53)(28,62,37,71)(29,63,38,31)(30,55,66,47)(34,41,64,42)(40,70,45,61)(48,65,59,68)(49,72,58,50);(1,4,5,6)(2,3,7,8)(9,13,14,11)(10,21,22,56)(12,32,33,27)(15,40,18,45)(16,38,17,29)(19,47,26,55)(20,51,52,69)(23,58,43,49)(24,48,25,59)(28,42,37,41)(30,60,66,46)(31,35,63,36)(34,71,64,62)(39,70,44,61)(50,67,72,57)(53,68,54,65)

Polycyclic presentation:

2^9|a^2=ef,b^2=ei,c^2=i,d^2=e,f^2=hi,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=def,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

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 417251);