#6816219

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
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

1749 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723415);