#9166764

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	#46298	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4096 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)(3,6)(4,8)(5,9)(7,10)(11,14)(12,13)(15,16)(17,18,22,24)(19,20,26,28)(21,23,32,31)(25,27,29,30)(34,36)(35,42)(38,46)(45,47);(1,3)(2,6)(4,11)(5,10)(7,9)(8,14)(12,15)(13,16)(17,19)(18,20)(21,29)(22,26)(23,30)(24,28)(25,32)(27,31)(33,34,39,36)(35,37,42,40)(38,41,46,44)(43,45,48,47);(3,10)(6,7)(11,16)(14,15)(17,20,22,28)(18,19,24,26)(21,27,32,30)(23,25,31,29)(33,35,39,42)(34,40,36,37)(38,43,46,48)(41,47,44,45);(1,3)(2,7)(4,11)(5,10)(6,9)(8,15)(12,14)(13,16)(17,20)(18,19)(21,30)(22,28)(23,29)(24,26)(25,31)(27,32)(33,36,39,34)(35,37,42,40)(38,44,46,41)(43,45,48,47);(1,2)(3,6)(4,12)(5,9)(7,10)(8,13)(11,15)(14,16)(17,21)(18,23)(19,25)(20,27)(22,32)(24,31)(26,29)(28,30)(33,37)(34,35)(36,42)(38,45)(39,40)(41,43)(44,48)(46,47);(1,4)(2,8)(3,11)(5,13)(6,14)(7,15)(9,12)(10,16)(17,18)(19,20)(21,31)(22,24)(23,32)(25,30)(26,28)(27,29)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9073960);