#6606158

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

2054 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)(13,14,18,22)(15,19,24,33)(16,26,30,43)(17,31,28,21)(20,35,37,29)(23,38,27,42)(25,41,39,34)(32,44,36,40);(2,4)(5,6,8,9)(7,10,11,12)(13,15,18,24)(14,19,22,33)(16,27,30,23)(17,25,28,39)(20,36,37,32)(21,34,31,41)(26,42,43,38)(29,44,35,40);(6,9)(7,11)(13,16,28,35,18,30,17,29)(14,20,31,43,22,37,21,26)(15,23,39,44,24,27,25,40)(19,32,41,42,33,36,34,38);(5,7)(6,10)(8,11)(9,12)(15,24)(19,33)(23,27)(25,39)(32,36)(34,41)(38,42)(40,44);(6,9)(7,11)(15,24)(19,33)(23,27)(25,39)(32,36)(34,41)(38,42)(40,44)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=gi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=di,e^b=eh,d^c=dhi,e^d=ehi

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, 6513354);