#9029737

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
prime radical	#57268	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3994 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)(3,9)(4,12)(5,6)(7,10)(8,13)(11,14)(15,16)(17,18,23,26)(19,24,22,20)(21,25,31,32)(27,29,30,28)(33,34)(35,41)(36,46)(37,43)(38,44)(39,40)(42,47)(45,48);(2,6)(7,9)(8,12)(14,15)(17,19,23,22)(18,24,26,20)(21,27,31,30)(25,29,32,28)(33,35,40,37)(34,41,39,43)(36,38,47,45)(42,48,46,44);(2,6)(3,10)(4,13)(14,15)(17,20,23,24)(18,19,26,22)(21,29,31,28)(25,30,32,27)(33,36,40,47)(34,42,39,46)(35,45,37,38)(41,44,43,48);(1,3)(2,7)(4,11)(5,10)(6,9)(8,14)(12,15)(13,16)(17,20)(18,19)(21,29)(22,26)(23,24)(25,30)(27,32)(28,31)(33,37,40,35)(34,43,39,41)(36,45,47,38)(42,44,46,48);(1,2)(3,9)(4,8)(5,6)(7,10)(11,15)(12,13)(14,16)(17,21)(18,25)(19,27)(20,28)(22,30)(23,31)(24,29)(26,32)(33,38,40,45)(34,44,39,48)(35,36,37,47)(41,46,43,42);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,22,23,19)(18,20,26,24)(21,30,31,27)(25,28,32,29)(33,39,40,34)(35,43,37,41)(36,42,47,46)(38,48,45,44)

Polycyclic presentation:

2^9|a^2=ghi,b^2=i,c^2=i,d^2=gh,e^2=gh,f^2=i,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, 8936933);