#6598375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573591
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×#550	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1851 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)(13,14,19,23)(15,20,27,36)(16,28,30,37)(17,22,18,31)(21,32,38,29)(24,39,42,43)(25,34,26,35)(33,40,44,41);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15,19,27)(14,20,23,36)(16,24,30,42)(17,26,18,25)(21,33,38,44)(22,35,31,34)(28,39,37,43)(29,41,32,40);(1,3,4,2)(6,9)(7,11)(13,16)(14,21)(15,24)(17,29)(18,32)(19,30)(20,33)(22,37)(23,38)(25,40)(26,41)(27,42)(28,31)(34,39)(35,43)(36,44);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,23,31)(15,25,27,26)(16,29,30,32)(20,34,36,35)(21,37,38,28)(24,40,42,41)(33,39,44,43);(6,9)(7,11)(13,17,19,18)(14,22,23,31)(15,26,27,25)(16,29,30,32)(20,35,36,34)(21,37,38,28)(24,41,42,40)(33,43,44,39)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,c^2=f,d^2=i,e^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dh,g^c=gi,e^d=eh

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505571);