#6598498

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×#555	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1826 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)(13,14,19,24)(15,20,27,35)(16,28,30,38)(17,23,18,22)(21,32,36,29)(25,39,41,43)(26,37,31,34)(33,40,44,42);(1,2,4,3)(5,6,8,9)(7,10,11,12)(13,15)(14,20)(16,25)(17,31)(18,26)(19,27)(21,33)(22,37)(23,34)(24,35)(28,39)(29,42)(30,41)(32,40)(36,44)(38,43);(1,3,4,2)(6,9)(7,11)(13,16,19,30)(14,21,24,36)(15,25,27,41)(17,29,18,32)(20,33,35,44)(22,28,23,38)(26,40,31,42)(34,43,37,39);(5,7)(6,10)(8,11)(9,12)(13,17,19,18)(14,22,24,23)(15,26,27,31)(16,29,30,32)(20,34,35,37)(21,28,36,38)(25,40,41,42)(33,43,44,39);(6,9)(7,11)(15,27)(20,35)(25,41)(26,31)(33,44)(34,37)(39,43)(40,42)

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fi,d^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,g^a=gi,c^b=ch,d^b=di,e^b=ehi,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, 6505694);