#6356388

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356301
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

1572 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:

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

Polycyclic presentation:

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

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, 6263584);