#6810610

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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
prime radical	#42646	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1847 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=f,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

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, 6717806);