#6596406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=hi,d^2=i,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,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, 6503602);