#9203899

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

4236 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)(3,8,10,4)(5,13,15,9)(11,12,16,14)(17,18,22,20)(19,23,26,24)(21,25,29,28)(27,30,32,31)(33,34,39,38)(35,43,46,40)(36,41,45,47)(37,42,44,48);(2,7)(4,8)(9,13)(12,14)(17,19)(18,23)(20,24)(21,27)(22,26)(25,30)(28,31)(29,32)(33,35)(34,40)(36,44)(37,45)(38,43)(39,46)(41,42)(47,48);(2,7)(3,10)(9,13)(11,16)(19,26)(23,24)(27,32)(30,31)(33,36,39,45)(34,41,38,47)(35,44,46,37)(40,42,43,48);(1,3,6,10)(2,8,7,4)(5,11,15,16)(9,14,13,12)(17,19)(18,24)(20,23)(21,27)(22,26)(25,31)(28,30)(29,32)(33,36)(34,41)(35,44)(37,46)(38,47)(39,45)(40,42)(43,48);(1,4,6,8)(2,3,7,10)(5,14,15,12)(9,16,13,11)(17,20,22,18)(19,24,26,23)(21,25,29,28)(27,30,32,31)(33,37,39,44)(34,42,38,48)(35,45,46,36)(40,41,43,47);(1,5,6,15)(2,9,7,13)(3,11,10,16)(4,12,8,14)(17,21)(18,25)(19,27)(20,28)(22,29)(23,30)(24,31)(26,32)(33,38,39,34)(35,43,46,40)(36,47,45,41)(37,42,44,48)

Polycyclic presentation:

2^9|a^2=gi,c^2=gh,d^2=h,e^2=gi,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9111095);