#7514905

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514172
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1897 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)(3,6)(4,5)(7,8)(9,10,15,19)(11,16,22,35)(12,24,27,29)(13,26,17,32)(14,30,33,18)(20,21,38,28)(23,37,31,36)(25,34,40,39);(2,5)(6,8)(9,11,15,22)(10,16,19,35)(12,25,27,40)(13,28,17,21)(14,31,33,23)(18,37,30,36)(20,32,38,26)(24,34,29,39);(1,3,4,7)(2,6,5,8)(9,12)(10,17)(11,20)(13,19)(14,32)(15,27)(16,34)(18,29)(21,37)(22,38)(23,40)(24,30)(25,31)(26,33)(28,36)(35,39);(9,13,15,17)(10,12,19,27)(11,21,22,28)(14,24,33,29)(16,25,35,40)(18,32,30,26)(20,37,38,36)(23,34,31,39);(2,5)(6,8)(11,22)(16,35)(20,38)(21,28)(23,31)(25,40)(34,39)(36,37)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=f,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cg,d^b=dh,e^b=ei,g^b=gi,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, 7422101);