#6356590

order	512
nilpotency class	3
isoclinism	isoclinic class of #6356533
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

1539 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6263786);