#6597200

order	512
nilpotency class	3
isoclinism	isoclinic class of #6574298
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×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2191 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)(5,6)(7,10)(8,11)(9,12)(13,14,18,22)(15,23,25,39)(16,20,28,36)(17,30,31,21)(19,32,34,26)(24,38,27,42)(29,44,40,37)(33,43,35,41);(5,7)(6,9)(8,12)(10,11)(13,15,17,26,18,25,31,32)(14,19,21,23,22,34,30,39)(16,27,29,43,28,24,40,41)(20,35,37,42,36,33,44,38);(2,4)(7,12)(9,10)(13,16)(14,20)(15,24)(17,29)(18,28)(19,33)(21,37)(22,36)(23,38)(25,27)(26,41)(30,44)(31,40)(32,43)(34,35)(39,42);(5,7)(6,10)(8,12)(9,11)(16,28)(20,36)(24,27)(29,40)(33,35)(37,44)(38,42)(41,43);(7,12)(9,10)(15,25)(19,34)(23,39)(24,27)(26,32)(33,35)(38,42)(41,43)

Polycyclic presentation:

2^9|a^2=i,b^2=fhi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,d^c=dh,e^d=ehi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6504396);