#6352107

order	512
nilpotency class	3
isoclinism	isoclinic class of #6352061
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	C₂²×C₄	C₂⁵
prime radical	#3449	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄×Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2741 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=bf,c^a=cg,f^a=fh,f^b=fh,g^b=gi,f^c=fi,e^d=ei

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, 6259303);