#6572634

order	512
nilpotency class	3
isoclinism	isoclinic class of #6533141
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂⁴	C₂⁵
prime radical	#23597	C₂
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

2820 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

(17,18)(19,28)(20,33,36,25)(21,37)(22,26)(23,27)(24,32)(29,41,45,38)(30,39)(31,42)(34,40,35,46)(43,47,44,48);(1,2,5,8)(3,9,11,6)(4,7,12,14)(10,15,16,13)(17,19)(18,24)(20,29)(21,30)(22,31)(23,32)(25,38)(26,39)(27,28)(33,41)(34,43)(35,44)(36,45)(37,42)(40,47)(46,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,20)(18,25)(19,29)(21,34)(22,35)(23,36)(24,38)(26,40)(27,33)(28,41)(30,43)(31,44)(32,45)(37,46)(39,47)(42,48);(1,4)(2,7)(3,10)(5,12)(6,13)(8,14)(9,15)(11,16)(17,21,23,22)(18,26,27,37)(19,30,32,31)(20,34,36,35)(24,39,28,42)(25,40,33,46)(29,43,45,44)(38,47,41,48);(4,12)(7,14)(10,16)(13,15)(17,22,23,21)(18,26,27,37)(19,31,32,30)(20,35,36,34)(24,39,28,42)(25,40,33,46)(29,44,45,43)(38,47,41,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6479830);