#9230861

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

4160 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)(15,16)(17,18)(19,26)(20,21)(22,23)(24,29)(25,32)(27,31)(28,30)(33,34,38,42)(35,39,41,37)(36,40,46,48)(43,45,44,47);(1,2,6,3)(4,7,12,9)(5,8,14,10)(11,15,13,16)(18,23)(20,25)(24,26)(27,30)(33,35,38,41)(34,39,42,37)(36,43,46,44)(40,45,48,47);(17,19)(18,24)(20,27)(21,28)(22,29)(23,26)(25,30)(31,32)(33,35)(34,37)(36,43)(38,41)(39,42)(40,47)(44,46)(45,48);(1,3,6,2)(4,7,12,9)(5,8,14,10)(11,16,13,15)(17,19)(18,24)(20,30)(21,31)(22,29)(23,26)(25,27)(28,32)(33,36,38,46)(34,40,42,48)(35,43,41,44)(37,47,39,45);(1,4,6,12)(2,7,3,9)(5,13,14,11)(8,16,10,15)(17,20)(18,21)(19,27)(22,25)(23,32)(24,28)(26,31)(29,30)(33,35)(34,39)(36,44)(37,42)(38,41)(40,47)(43,46)(45,48);(1,5,6,14)(2,8,3,10)(4,11,12,13)(7,15,9,16)(17,21)(18,25)(19,28)(20,23)(22,32)(24,30)(26,27)(29,31)(33,37,38,39)(34,41,42,35)(36,45,46,47)(40,43,48,44)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=gh,e^2=ghi,f^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=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, 9138057);