#10247407

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

4099 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,d^2=i,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, 10154603);