#10482485

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
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3849 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:

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

Polycyclic presentation:

2^9|a^2=gh,c^2=h,d^2=gi,e^2=g,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, 10389681);