#408622

order	512
nilpotency class	3
isoclinism	isoclinic class of #395173
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2408
prime radical	#60453	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1308 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)(3,8)(4,6)(5,7)(9,10)(11,22)(12,26)(13,28)(14,21)(15,19)(16,20)(17,24)(18,30)(23,37)(25,33)(27,32)(29,35)(31,34)(36,39)(38,40);(9,11,15,25,16,24,14,12)(10,17,21,33,20,22,19,26)(13,23,30,39,31,38,29,27)(18,32,28,40,35,36,34,37);(1,3,5,4)(2,6,7,8)(9,12,14,24,16,25,15,11)(10,17,19,26,20,22,21,33)(13,27,29,38,31,39,30,23)(18,32,34,37,35,36,28,40);(1,4,5,3)(2,6,7,8)(9,13,14,29,16,31,15,30)(10,18,19,34,20,35,21,28)(11,23,12,27,24,38,25,39)(17,32,26,37,22,36,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 315818);