#6578081

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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 ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2102 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:

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

Polycyclic presentation:

2^9|c^2=i,d^2=fi,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=ei,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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6485277);