#6566141

order	512
nilpotency class	3
isoclinism	isoclinic class of #6535696
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#3475	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×D₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1588 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=f,d^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=di,e^b=ei,g^c=gi,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, 6473337);