#6584884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573451
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×#3307	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1914 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)(8,12)(9,10)(13,14,17,20)(15,19,26,33)(16,21,28,34)(18,22,30,35)(23,38,25,31)(24,32,40,43)(27,37,29,36)(39,44,41,42);(6,10)(11,12)(13,15)(14,19)(16,24)(17,26)(18,23)(20,33)(21,32)(22,38)(25,30)(27,39)(28,40)(29,41)(31,35)(34,43)(36,42)(37,44);(1,2,3,4)(5,7,9,8)(6,11,10,12)(14,20)(15,23)(16,27)(18,30)(19,31)(21,36)(25,26)(28,29)(32,43)(33,38)(34,37)(39,41);(5,8,9,7)(6,12,10,11)(13,16,17,28)(14,21,20,34)(15,24,26,40)(18,27,30,29)(19,32,33,43)(22,37,35,36)(23,39,25,41)(31,42,38,44);(15,25)(16,27)(18,30)(19,31)(21,37)(22,35)(23,26)(24,40)(28,29)(32,43)(33,38)(34,36)

Polycyclic presentation:

2^9|a^2=i,c^2=fgi,d^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,e^b=ehi,h^b=hi,d^c=dhi,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492080);