#7503926

order	512
nilpotency class	3
isoclinism	isoclinic class of #7500908
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

1808 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=f,c^2=i,d^2=f,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dfg,e^a=ei,c^b=ch,d^b=df,e^b=ei,h^b=hi,g^c=gi,h^c=hi,g^d=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7411122);