#6570612

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535815
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

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 ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2207 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)(5,6,11,10)(7,8,9,12)(13,14)(15,23)(16,20)(17,21)(18,28)(19,22)(24,27)(25,26);(5,7,11,9)(6,12,10,8)(13,15,18,20,19,25,17,24)(14,16,21,23,22,27,28,26);(1,2,3,4)(5,8,11,12)(6,7,10,9)(13,16,19,27)(14,20,22,24)(15,21,25,28)(17,26,18,23);(5,9,11,7)(6,12,10,8)(13,17,19,18)(14,21,22,28)(15,24,25,20)(16,23,27,26);(5,10,11,6)(7,8,9,12)(13,17,19,18)(14,21,22,28)(15,20,25,24)(16,26,27,23)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6477808);