#6816323

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

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=gh,d^2=i,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, 6723519);