#6816321

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682460
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	C₂×#469	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1755 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=ghi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723517);