#7561750

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

2224 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=fgi,e^2=i,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,e^d=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, 7468946);