#6465787

order	512
nilpotency class	3
isoclinism	isoclinic class of #25443
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂×D₄²
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1690 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=gh,d^2=h,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,h^b=hi,d^c=di

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, 6372983);