#10455709

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

4009 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)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,18,23,27)(19,22,21,24)(20,26,31,25)(28,32,29,30)(36,44)(39,45)(42,48)(46,47);(2,7)(8,9)(10,14)(15,16)(17,19,23,21)(18,24,27,22)(20,29,31,28)(25,30,26,32)(33,34)(35,40)(36,45)(37,41)(38,43)(39,44)(42,46)(47,48);(1,3,6,4)(2,8,7,9)(5,11,13,12)(10,15,14,16)(17,20)(18,25)(19,28)(21,29)(22,30)(23,31)(24,32)(26,27)(33,35)(34,38)(36,42)(37,43)(39,46)(40,41)(44,48)(45,47);(1,2)(3,8)(4,9)(5,10)(6,7)(11,15)(12,16)(13,14)(17,21,23,19)(18,22,27,24)(20,29,31,28)(25,30,26,32)(33,36)(34,39)(35,42)(37,44)(38,46)(40,47)(41,45)(43,48);(1,4,6,3)(2,9,7,8)(5,11,13,12)(10,15,14,16)(17,20)(18,26)(19,29)(21,28)(22,30)(23,31)(24,32)(25,27)(33,35)(34,40)(36,42)(37,43)(38,41)(39,47)(44,48)(45,46);(1,5,6,13)(2,10,7,14)(3,11,4,12)(8,15,9,16)(17,22)(18,19)(20,30)(21,27)(23,24)(25,28)(26,29)(31,32)(33,34)(35,38)(36,39)(37,41)(40,43)(42,46)(44,45)(47,48)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gh,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,d^b=dgi,e^d=eh,f^e=fgi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10362905);