#9203645

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	#54011	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4081 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)(3,7)(4,5)(6,9)(8,10)(11,12)(13,16)(14,15)(17,18)(19,26)(20,25)(21,23)(22,24)(27,32)(28,31)(29,30)(33,34,40,44)(35,45,37,41)(36,42,46,47)(38,39,43,48);(1,3)(2,7)(4,11)(5,12)(6,10)(8,9)(13,14)(15,16)(17,19,22,21)(18,23,24,26)(20,28,30,27)(25,32,29,31)(33,35,40,37)(34,41,44,45)(36,38,46,43)(39,42,48,47);(3,10)(7,8)(11,13)(12,16)(18,24)(23,26)(25,29)(31,32)(33,36,40,46)(34,42,44,47)(35,38,37,43)(39,41,48,45);(1,3)(2,8)(4,13)(5,12)(6,10)(7,9)(11,14)(15,16)(17,18)(19,23)(20,29)(21,26)(22,24)(25,30)(27,32)(28,31)(33,37,40,35)(34,41,44,45)(36,38,46,43)(39,47,48,42);(1,4,6,14)(2,5,9,15)(3,11,10,13)(7,12,8,16)(17,20,22,30)(18,25,24,29)(19,27,21,28)(23,31,26,32)(33,38)(34,39)(35,46)(36,37)(40,43)(41,42)(44,48)(45,47);(1,5)(2,4)(3,12)(6,15)(7,11)(8,13)(9,14)(10,16)(17,21,22,19)(18,26,24,23)(20,27,30,28)(25,31,29,32)(33,39)(34,43)(35,42)(36,41)(37,47)(38,44)(40,48)(45,46)

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=gh,e^2=hi,f^2=h,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, 9110841);