#6355931

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
prime radical	D₄×#41	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1553 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(17,18,23,27)(19,28,29,42)(20,25,34,41)(21,26,22,36)(24,31,37,32)(30,43,44,48)(33,39,35,40)(38,46,47,45);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(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);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(19,29)(24,37)(28,42)(30,44)(31,32)(38,47)(43,48)(45,46);(3,10)(6,14)(11,16)(13,15)(17,20)(18,25)(19,30)(21,35)(22,33)(23,34)(24,38)(26,40)(27,41)(28,43)(29,44)(31,46)(32,45)(36,39)(37,47)(42,48);(2,8)(6,14)(7,9)(13,15)(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|a^2=h,b^2=fh,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, 6263127);