#7560615

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
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

1945 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:

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

Polycyclic presentation:

2^9|c^2=fg,d^2=g,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=efi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7467811);