#10196033

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

4201 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=i,e^2=g,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dh,e^d=egi,f^e=fgi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 10103229);