#6810622

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	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1754 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(6,10)(7,8)(13,14,19,18)(15,20,25,33)(16,21,27,29)(17,22,32,31)(23,26,34,38)(24,30,36,43)(28,35,40,42)(37,39,41,44);(5,6)(7,12)(8,11)(9,10)(13,15,19,25)(14,20,18,33)(16,26,27,38)(17,30,32,43)(21,34,29,23)(22,36,31,24)(28,41,40,37)(35,44,42,39);(2,4)(5,7,9,8)(6,11,10,12)(13,16,14,21,19,27,18,29)(15,23,33,38,25,34,20,26)(17,28,22,35,32,40,31,42)(24,37,43,44,36,41,30,39);(5,8,9,7)(6,11,10,12)(16,27)(21,29)(23,34)(26,38)(28,40)(35,42)(37,41)(39,44);(13,17,14,22,19,32,18,31)(15,24,33,43,25,36,20,30)(16,28,21,35,27,40,29,42)(23,37,38,44,34,41,26,39)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=f,e^2=fhi,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, 6717818);