#10312379

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

3987 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=h,d^2=h,e^2=gh,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dgi,e^b=eghi,d^c=dghi,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, 10219575);