#7521892

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 ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1989 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=f,e^2=fi,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, 7429088);