#10247347

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

4135 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)(3,10)(4,5)(6,8)(7,14)(9,11)(12,13)(15,16)(17,18,23,21)(19,24,25,20)(22,29,30,26)(27,28,32,31)(33,34,39,40)(35,41,37,43)(36,46,47,42)(38,44,48,45);(1,3,6,14)(2,7,8,10)(4,15,11,12)(5,13,9,16)(17,19)(18,24)(20,21)(22,27)(23,25)(26,31)(28,29)(30,32)(34,40)(36,47)(41,43)(44,45);(2,8)(4,11)(7,10)(12,15)(17,20,23,24)(18,25,21,19)(22,28,30,31)(26,32,29,27)(33,35,39,37)(34,41,40,43)(36,45,47,44)(38,42,48,46);(1,4,6,11)(2,9,8,5)(3,12,14,15)(7,13,10,16)(17,20)(18,19)(21,25)(22,28)(23,24)(26,27)(29,32)(30,31)(33,36)(34,42)(35,45)(37,44)(38,43)(39,47)(40,46)(41,48);(1,3)(2,10)(4,15)(5,16)(6,14)(7,8)(9,13)(11,12)(17,21,23,18)(19,24,25,20)(22,26,30,29)(27,28,32,31)(33,37,39,35)(34,43,40,41)(36,45,47,44)(38,42,48,46);(1,5,6,9)(2,11,8,4)(3,13,14,16)(7,12,10,15)(17,22)(18,26)(19,27)(20,28)(21,29)(23,30)(24,31)(25,32)(33,38,39,48)(34,44,40,45)(35,42,37,46)(36,43,47,41)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=hi,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, 10154543);