#6554683

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533123
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×#127	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1749 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=hi,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,d^b=di,g^c=gi,e^d=eh

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6461879);