#6403961

order	512
nilpotency class	3
isoclinism	isoclinic class of #62307
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂³×D₄
commutator	C₂×C₄	C₂⁴×C₄
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	#61141	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1765 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=fg,e^2=h,g^2=i,h^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ci,d^b=df,e^b=ei,e^c=ei,e^d=ei

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, 6311157);