#6396567

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	#60797	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#433 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1813 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,e^2=f,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^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, 6303763);