#9005153

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

4138 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,16)(14,15)(33,34,39,43)(35,40,38,48)(36,37,42,41)(44,46,45,47);(2,7)(4,15)(8,11)(10,13)(17,18,22,20)(19,23,25,27)(21,24,29,28)(26,30,31,32)(33,35,39,38)(34,40,43,48)(36,45,42,44)(37,47,41,46);(2,7)(3,12)(4,15)(5,16)(18,20)(19,25)(21,29)(30,32)(33,36,39,42)(34,41,43,37)(35,44,38,45)(40,47,48,46);(1,3,6,12)(2,8,7,11)(4,10,15,13)(5,14,16,9)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36)(34,37)(35,45)(38,44)(39,42)(40,47)(41,43)(46,48);(1,4,6,15)(2,9,7,14)(3,13,12,10)(5,8,16,11)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,37)(34,42)(35,46)(36,43)(38,47)(39,41)(40,45)(44,48);(1,5,6,16)(2,10,7,13)(3,14,12,9)(4,8,15,11)(17,21)(18,24)(19,26)(20,28)(22,29)(23,30)(25,31)(27,32)(33,38,39,35)(34,40,43,48)(36,44,42,45)(37,47,41,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=gh,d^2=g,e^2=hi,f^2=h,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,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, 8912349);