#7516556

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

Polycyclic presentation:

2^9|b^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,e^c=ei,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, 7423752);