#7148223

order	512
nilpotency class	3
isoclinism	isoclinic class of #6993947
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

1921 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,e^2=h,g^2=i,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,d^b=df,e^b=eh,d^c=dh,g^c=gi,e^d=ei

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7055419);