#6815843

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

2209 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,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, 6723039);