#9065948

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

4170 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8973144);