#417330

order	512
nilpotency class	3
isoclinism	isoclinic class of #397902
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2397
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1028 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 324526);