#6355963

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355925
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

1503 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)(4,8)(5,10)(6,11)(7,15)(9,14)(13,16)(17,18)(19,28)(20,25)(21,26)(22,36)(23,27)(24,32)(29,42)(30,43)(31,37)(33,39)(34,41)(35,40)(38,45)(44,48)(46,47);(1,3,6,4)(2,8,11,12)(5,14,7,13)(9,10,16,15)(17,19,21,31,23,29,22,32)(18,24,36,42,27,37,26,28)(20,30,35,46,34,44,33,45)(25,38,39,48,41,47,40,43);(1,4,6,3)(2,9,11,16)(5,14,7,13)(8,10,12,15)(19,29)(24,37)(28,42)(30,44)(31,32)(38,47)(43,48)(45,46);(1,5,6,7)(2,10,11,15)(3,13,4,14)(8,9,12,16)(17,20,23,34)(18,25,27,41)(19,30,29,44)(21,35,22,33)(24,38,37,47)(26,40,36,39)(28,43,42,48)(31,46,32,45);(2,11)(8,12)(9,16)(10,15)(17,21,23,22)(18,26,27,36)(19,31,29,32)(20,33,34,35)(24,28,37,42)(25,39,41,40)(30,45,44,46)(38,48,47,43)

Polycyclic presentation:

2^9|b^2=fh,c^2=i,d^2=hi,e^2=h,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ei,f^a=fh,g^a=gi,c^b=ch,d^b=di,g^c=gi,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, 6263159);