#6598465

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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×#548	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1830 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)(13,14,19,22)(15,20,26,36)(16,29,31,38)(17,24,18,23)(21,32,37,30)(25,39,40,44)(27,35,28,34)(33,42,43,41);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,25)(17,27)(18,28)(19,26)(21,33)(22,36)(23,34)(24,35)(29,39)(30,41)(31,40)(32,42)(37,43)(38,44);(1,3,4,2)(6,9)(7,11)(13,16,19,31)(14,21,22,37)(15,25,26,40)(17,30,18,32)(20,33,36,43)(23,29,24,38)(27,41,28,42)(34,39,35,44);(5,7)(6,10)(8,11)(9,12)(14,22)(15,26)(21,37)(23,24)(25,40)(27,28)(29,38)(41,42);(6,9)(7,11)(13,17,19,18)(14,23,22,24)(15,27,26,28)(16,30,31,32)(20,34,36,35)(21,29,37,38)(25,41,40,42)(33,39,43,44)

Polycyclic presentation:

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

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6505661);