#6357469

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357459
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	#60782	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1861 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264665);