#6577613

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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 ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2112 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)(7,11)(8,10)(9,12)(13,14)(15,22)(16,20)(17,21)(18,26)(19,24)(23,27)(25,28);(1,2)(3,4)(5,7,9,8)(6,10,12,11)(13,14,17,21)(15,23,25,19)(16,26,18,20)(22,24,28,27);(2,4)(5,7,9,8)(6,11,12,10)(13,15,17,25)(14,19,21,23)(16,24,18,27)(20,28,26,22);(1,2)(3,4)(5,8,9,7)(6,10,12,11)(13,14)(15,23)(16,26)(17,21)(18,20)(19,25)(22,27)(24,28);(2,4)(13,16,17,18)(14,20,21,26)(15,24,25,27)(19,28,23,22)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6484809);