#9723156

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

4248 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=gi,d^2=g,e^2=g,f^2=i,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^e=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9630352);