#10407757

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4263 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=g,d^2=gi,e^2=hi,f^2=hi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,f^b=fgi,e^c=egi,e^d=eghi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10314953);