#7613564

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

2139 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)(13,14,20,22)(15,18,16,19)(17,24,27,23)(21,25,28,26);(2,4)(6,9)(10,12)(13,15)(14,18)(16,20)(17,21)(19,22)(23,26)(24,25)(27,28);(5,7,8,11)(6,10,9,12)(13,16,19,22,20,15,18,14)(17,23,26,21,27,24,25,28);(1,2,3,4)(13,17,19,26,20,27,18,25)(14,21,16,24,22,28,15,23);(6,9)(10,12)(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=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, 7520760);