#10247323

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

4234 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)(3,9)(4,8)(5,10)(6,7)(11,16)(12,13)(14,15)(17,18)(19,27)(20,23)(21,25)(22,30)(24,28)(26,31)(29,32)(33,34,39,37)(35,40,41,36)(38,45,46,42)(43,44,48,47);(1,3)(2,5)(4,12)(6,10)(7,9)(8,16)(11,14)(13,15)(17,19,23,28)(18,24,20,27)(21,31,32,22)(25,30,29,26)(33,35)(34,40)(36,37)(38,43)(39,41)(42,47)(44,45)(46,48);(2,7)(3,10)(8,15)(11,12)(18,20)(24,27)(25,29)(26,30)(33,36,39,40)(34,41,37,35)(38,44,46,47)(42,48,45,43);(1,3)(2,5)(4,11)(6,10)(7,9)(8,13)(12,14)(15,16)(17,20,23,18)(19,24,28,27)(21,25,32,29)(22,30,31,26)(33,36)(34,35)(37,41)(38,44)(39,40)(42,43)(45,48)(46,47);(1,4,6,14)(2,8,7,15)(3,11,10,12)(5,13,9,16)(17,21,23,32)(18,25,20,29)(19,22,28,31)(24,26,27,30)(33,37,39,34)(35,40,41,36)(38,42,46,45)(43,44,48,47);(1,5)(2,3)(4,13)(6,9)(7,10)(8,11)(12,15)(14,16)(17,22)(18,26)(19,21)(20,30)(23,31)(24,25)(27,29)(28,32)(33,38)(34,42)(35,43)(36,44)(37,45)(39,46)(40,47)(41,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=gh,d^2=hi,e^2=h,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, 10154519);