#6584913

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1820 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)(15,23,26,19)(16,28,31,20)(18,22,32,38)(24,35,27,33)(25,34,41,39)(29,37,30,36)(40,43,42,44);(6,10)(11,12)(13,15,17,26)(14,19,21,23)(16,25,31,41)(18,24,32,27)(20,34,28,39)(22,33,38,35)(29,42,30,40)(36,44,37,43);(1,2,3,4)(15,24)(16,29)(18,32)(19,33)(20,36)(22,38)(23,35)(25,41)(26,27)(28,37)(30,31)(34,39);(5,7,9,8)(6,11,10,12)(13,16)(14,20)(15,25)(17,31)(18,30)(19,34)(21,28)(22,37)(23,39)(24,40)(26,41)(27,42)(29,32)(33,43)(35,44)(36,38);(5,8,9,7)(6,12,10,11)(15,24)(16,30)(18,32)(19,33)(20,37)(22,38)(23,35)(26,27)(28,36)(29,31)(40,42)(43,44)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=g,d^2=fi,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=eh,h^b=hi,d^c=dh,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, 6492109);