#10247493

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

4192 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=g,d^2=i,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, 10154689);