#6357637

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357620
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1916 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)(4,10)(6,12)(8,13)(11,15)(14,16)(17,18,22,26)(19,27,28,40)(20,24,31,38)(21,33,34,25)(23,35,36,30)(29,41,42,48)(32,45,44,39)(37,46,47,43);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,21,30,22,28,34,35)(18,23,25,27,26,36,33,40)(20,29,32,43,31,42,44,46)(24,37,39,41,38,47,45,48);(1,3,10,15,5,11,4,9)(2,6,13,16,7,14,8,12)(19,28)(23,36)(27,40)(29,42)(30,35)(37,47)(41,48)(43,46);(2,7)(6,14)(8,13)(12,16)(17,20)(18,24)(19,29)(21,32)(22,31)(23,37)(25,39)(26,38)(27,41)(28,42)(30,43)(33,45)(34,44)(35,46)(36,47)(40,48);(20,31)(24,38)(29,42)(32,44)(37,47)(39,45)(41,48)(43,46)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=gi,f^2=h,g^2=i,b^a=bf,c^a=cg,f^a=fh,g^a=gi,c^b=ch,d^b=di,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, 6264833);