#6358090

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358057
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

1792 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265286);