#9203712

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	#53523	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4087 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,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, 9110908);