#6584971

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573511
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×#3339	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1884 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)(13,14,17,21)(15,23,27,19)(16,20,30,38)(18,22,32,40)(24,35,26,33)(25,36,28,34)(29,37,31,39)(41,44,42,43);(6,10)(11,12)(13,15)(14,19)(16,28)(17,27)(18,24)(20,36)(21,23)(22,33)(25,30)(26,32)(29,42)(31,41)(34,38)(35,40)(37,44)(39,43);(1,2,3,4)(15,24)(16,29)(18,32)(19,33)(20,37)(22,40)(23,35)(26,27)(30,31)(38,39)(41,42)(43,44);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,25)(17,30)(18,29)(19,34)(21,38)(22,37)(23,36)(24,41)(26,42)(27,28)(31,32)(33,43)(35,44)(39,40);(5,8,9,7)(6,12,10,11)(15,26)(16,29)(18,32)(19,35)(20,37)(22,40)(23,33)(24,27)(25,28)(30,31)(34,36)(38,39)

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=di,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

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6492167);