#7613565

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518964
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

2159 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)(14,20)(15,24)(16,21)(17,19)(22,25)(26,27);(2,4)(6,9)(10,12)(13,14)(15,16)(17,23)(18,20)(19,28)(21,22)(24,26)(25,27);(5,7,8,11)(6,10,9,12)(13,15,19,24,18,25,17,22)(14,21,23,27,20,26,28,16);(1,2,3,4)(13,16,19,21,18,27,17,26)(14,22,23,15,20,24,28,25);(6,9)(10,12)(13,17,18,19)(14,23,20,28)(15,24,25,22)(16,26,27,21)

Polycyclic presentation:

2^9|c^2=fgi,d^2=h,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,d^c=di,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, 7520761);