#9206546

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	#52748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3956 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)(3,8)(4,12)(5,7)(6,11)(9,14)(10,15)(13,16)(17,18,22,23)(19,25,26,20)(21,29,31,24)(27,28,32,30)(34,39)(37,40)(41,46)(45,47);(1,3,5,11)(2,6,7,8)(4,10,14,13)(9,15,12,16)(17,18,22,23)(19,20,26,25)(21,24,31,29)(27,28,32,30)(36,44)(41,46)(43,48)(45,47);(2,7)(6,8)(9,12)(15,16)(18,23)(19,26)(24,29)(27,32)(33,34)(35,40)(36,41)(37,42)(38,39)(43,47)(44,46)(45,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,15)(12,14)(13,16)(17,19,22,26)(18,20,23,25)(21,27,31,32)(24,28,29,30)(33,35,38,42)(34,40,39,37)(36,43,44,48)(41,47,46,45);(1,3)(2,8)(4,13)(5,11)(6,7)(9,15)(10,14)(12,16)(17,20)(18,19)(21,30)(22,25)(23,26)(24,32)(27,29)(28,31)(33,36,38,44)(34,41,39,46)(35,43,42,48)(37,45,40,47);(1,4)(2,9)(3,10)(5,14)(6,15)(7,12)(8,16)(11,13)(17,21,22,31)(18,24,23,29)(19,27,26,32)(20,28,25,30)(33,37,38,40)(34,42,39,35)(36,45,44,47)(41,48,46,43)

Polycyclic presentation:

2^9|a^2=i,b^2=gi,d^2=gh,e^2=ghi,f^2=gh,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, 9113742);