#416367

order	512
nilpotency class	3
isoclinism	isoclinic class of #395944
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

972 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efi,b^2=eh,c^2=e,d^2=h,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,g^b=gi,f^c=fi,g^c=gi

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, 323563);