#10247063

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

4002 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)(3,6)(4,7)(5,10)(8,14)(9,11)(12,16)(13,15)(17,18,24,27)(19,23,22,25)(20,26,21,31)(28,30,29,32)(33,34,38,41)(35,42,37,39)(36,40,44,47)(43,48,45,46);(2,3)(4,11)(8,10)(12,13)(17,19,24,22)(18,25,27,23)(20,28,21,29)(26,32,31,30)(35,37)(36,44)(39,42)(40,47);(2,3)(7,9)(8,10)(15,16)(17,20,24,21)(18,26,27,31)(19,28,22,29)(23,30,25,32)(33,35,38,37)(34,39,41,42)(36,45,44,43)(40,48,47,46);(1,3,6,2)(4,7,11,9)(5,10,14,8)(12,15,13,16)(17,21,24,20)(18,26,27,31)(19,28,22,29)(23,32,25,30)(33,36,38,44)(34,40,41,47)(35,43,37,45)(39,46,42,48);(1,4)(2,7)(3,9)(5,13)(6,11)(8,16)(10,15)(12,14)(17,22,24,19)(18,25,27,23)(20,29,21,28)(26,32,31,30)(33,37,38,35)(34,39,41,42)(36,43,44,45)(40,48,47,46);(1,5,6,14)(2,8,3,10)(4,12,11,13)(7,15,9,16)(17,23,24,25)(18,22,27,19)(20,30,21,32)(26,29,31,28)(33,34)(35,39)(36,40)(37,42)(38,41)(43,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=gi,e^2=gh,f^2=g,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, 10154259);