#6606406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574024
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×#121	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1769 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,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,e^d=eh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6513602);