#423822

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1317 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=f,d^2=ei,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 331018);