#6356767

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356733
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	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1577 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)(3,8)(4,9)(5,14)(6,10)(7,11)(12,15)(13,16)(17,18,22,26)(19,27,31,43)(20,24,33,40)(21,35,28,25)(23,36,38,30)(29,42,32,46)(34,48,44,41)(37,47,39,45);(17,19,28,36,22,31,21,30)(18,23,35,43,26,38,25,27)(20,32,44,45,33,29,34,47)(24,39,48,42,40,37,41,46);(1,3)(2,7)(4,11)(5,13)(6,12)(8,14)(9,15)(10,16);(1,3)(2,8)(4,13)(5,11)(6,12)(7,14)(9,16)(10,15)(17,20)(18,24)(19,29)(21,34)(22,33)(23,37)(25,41)(26,40)(27,42)(28,44)(30,45)(31,32)(35,48)(36,47)(38,39)(43,46);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,15,16,8)(20,33)(24,40)(29,32)(34,44)(37,39)(41,48)(42,46)(45,47)

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263963);