#10247673

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

3989 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=hi,e^2=g,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,d^b=dgi,e^b=eh,d^c=dh,e^d=egi,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, 10154869);