#7562170

order	512
nilpotency class	3
isoclinism	isoclinic class of #7519008
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2358 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 28 points:

(5,6)(7,11)(8,12)(9,10)(13,14,20,22)(15,18,16,19)(17,23,27,24)(21,26,28,25);(1,2)(3,4)(6,10)(11,12)(13,15)(14,18)(16,20)(17,21)(19,22)(23,26)(24,25)(27,28);(5,7,9,8)(6,11,10,12)(13,16,18,14,20,15,19,22)(17,24,25,21,27,23,26,28);(2,4)(5,8,9,7)(6,12,10,11)(13,17,19,26,20,27,18,25)(14,21,16,24,22,28,15,23);(6,10)(11,12)(13,18,20,19)(14,16,22,15)(17,25,27,26)(21,24,28,23)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,d^2=fgi,e^2=i,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,e^c=ei

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7469366);