#6506736

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500456
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3004 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)(13,14,19,23)(15,24,27,34)(16,21,28,36)(17,22,18,32)(20,30,35,26)(25,39,40,44)(29,37,31,38)(33,41,43,42);(5,6)(7,11)(8,10)(9,12)(13,15,17,30,19,27,18,26)(14,20,32,34,23,35,22,24)(16,25,31,41,28,40,29,42)(21,33,37,44,36,43,38,39);(1,2,3,4)(5,7)(6,9)(8,12)(10,11)(16,28)(21,36)(25,40)(29,31)(33,43)(37,38)(39,44)(41,42);(7,12)(9,11)(13,16)(14,21)(15,25)(17,31)(18,29)(19,28)(20,33)(22,38)(23,36)(24,39)(26,42)(27,40)(30,41)(32,37)(34,44)(35,43);(6,10)(9,11)(13,17,19,18)(14,22,23,32)(15,26,27,30)(16,29,28,31)(20,34,35,24)(21,37,36,38)(25,41,40,42)(33,39,43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=fi,c^2=g,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=ehi,d^c=dhi,e^d=ei

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, 6413932);