#6810249

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
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
prime radical	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1991 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)(5,6)(7,11)(8,10)(9,12);(5,7,8,12)(6,9,10,11)(13,14,18,23)(15,24,29,35)(16,20,25,22)(17,31,32,36)(19,26,34,28)(21,37,38,30)(27,41,43,33)(39,40,44,42);(2,4)(7,12)(9,11)(13,15,25,26,18,29,16,28)(14,19,20,35,23,34,22,24)(17,27,37,40,32,43,30,42)(21,33,36,44,38,41,31,39);(6,10)(9,11)(13,16,18,25)(14,20,23,22)(15,26,29,28)(17,30,32,37)(19,24,34,35)(21,36,38,31)(27,40,43,42)(33,39,41,44);(13,17,25,37,18,32,16,30)(14,21,20,36,23,38,22,31)(15,27,26,40,29,43,28,42)(19,33,35,44,34,41,24,39)

Polycyclic presentation:

2^9|b^2=fi,c^2=fhi,d^2=i,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, 6717445);