#8944429

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

4137 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:

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

Polycyclic presentation:

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

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, 8851625);