#6520119

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500120
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×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2285 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1753 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)(13,14,18,24,19,22,17,23)(15,25,28,35,29,36,27,20)(16,21,32,38,31,39,33,40)(26,41,42,37,30,44,43,34);(5,6)(7,10)(8,9)(11,12)(13,15,19,29)(14,20,22,35)(16,30,31,26)(17,28,18,27)(21,37,39,34)(23,25,24,36)(32,42,33,43)(38,41,40,44);(1,2,3,4)(6,9)(7,11)(13,16)(14,21)(15,26)(17,33)(18,32)(19,31)(20,34)(22,39)(23,40)(24,38)(25,41)(27,43)(28,42)(29,30)(35,37)(36,44);(5,7,8,11)(6,10,9,12)(14,22)(16,31)(20,35)(23,24)(25,36)(26,30)(32,33)(42,43);(6,9)(10,12)(13,17,19,18)(14,23,22,24)(15,27,29,28)(16,32,31,33)(20,36,35,25)(21,38,39,40)(26,42,30,43)(34,41,37,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6427315);