#6348012

order	512
nilpotency class	3
isoclinism	isoclinic class of #6347957
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#120
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

2737 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255208);