#8733934

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

4143 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=gh,d^2=h,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,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, 8641130);