#10480955

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

4332 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=gh,d^2=i,f^2=g,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, 10388151);