#7613707

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 ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2269 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)(7,10)(8,9)(11,12)(14,19)(15,23)(16,17)(20,25)(21,22)(24,26);(2,4)(6,9)(10,12)(14,19)(15,24)(16,17)(21,22)(23,26)(27,28);(5,7,8,11)(6,10,9,12)(13,14,17,21,18,22,16,19)(15,25,24,27,26,20,23,28);(1,2,3,4)(13,15,16,23,18,26,17,24)(14,20,19,27,22,25,21,28);(6,9)(10,12)(15,26)(20,25)(23,24)(27,28)

Polycyclic presentation:

2^9|c^2=fgi,d^2=hi,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=cg,d^b=dh,g^b=gi,h^b=hi,d^c=di,e^d=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, 7520903);