#507233

order	512
nilpotency class	4
isoclinism	isoclinic class of #500829
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#13304	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊C₄ ◁ C₈×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

565 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,34)(28,44)(29,36)(30,37)(31,38)(32,39)(33,41)(35,45)(40,43)(42,47)(46,48);(1,3,4,8)(2,5,6,7)(10,13)(14,15)(17,19,27,42,32,46,33,28,20,29,21,30,22,31,43,45)(18,23,34,47,39,48,41,44,24,36,25,37,26,38,40,35);(10,13)(14,15)(18,24)(19,28)(21,33)(22,32)(23,35)(25,40)(27,43)(29,45)(30,31)(34,41)(36,44)(37,48)(38,47)(42,46);(2,6)(5,7)(9,11,12,16)(10,14,13,15)

Polycyclic presentation:

2^9|b^2=efg,d^2=fi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,g^c=gh,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 414429);