#7516528

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

1948 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)(11,19)(12,24)(13,27)(14,17)(15,30)(16,22)(18,29)(20,39)(21,40)(23,36)(25,38)(26,33)(28,32)(31,37)(34,35);(2,5)(6,8)(9,11,14,22)(10,16,17,19)(12,25,26,35)(13,21,28,39)(15,31,29,23)(18,37,30,36)(20,32,40,27)(24,34,33,38);(9,12,14,26)(10,13,17,28)(11,20,22,40)(15,32,29,27)(16,34,19,38)(18,33,30,24)(21,36,39,37)(23,35,31,25);(1,3,4,7)(2,6,5,8)(9,13,14,28)(10,12,17,26)(11,21,22,39)(15,33,29,24)(16,35,19,25)(18,32,30,27)(20,36,40,37)(23,34,31,38);(2,5)(6,8)(10,17)(13,28)(16,19)(18,30)(21,39)(24,33)(34,38)(36,37)

Polycyclic presentation:

2^9|b^2=i,c^2=i,d^2=f,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,g^b=gi,g^c=gi,h^c=hi,e^d=ei,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, 7423724);