#9210288

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	#2661	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3935 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)(3,8)(4,7)(5,9)(6,11)(10,16)(12,15)(13,14)(17,18,23,22)(19,24,27,28)(20,29,30,25)(21,26,31,32)(33,34)(35,42)(36,41)(37,40)(38,39)(43,48)(44,47)(45,46);(1,3)(2,6)(4,10)(5,11)(7,14)(8,9)(12,13)(15,16)(17,19)(18,24)(20,21)(22,28)(23,27)(25,32)(26,29)(30,31)(33,34,39,38)(35,40,37,42)(36,41,46,45)(43,47,44,48);(4,12)(7,15)(10,13)(14,16)(19,27)(20,30)(24,28)(25,29)(33,35,39,37)(34,40,38,42)(36,44,46,43)(41,48,45,47);(1,4,5,12)(2,7,9,15)(3,10,11,13)(6,14,8,16)(17,20,23,30)(18,25,22,29)(19,21,27,31)(24,32,28,26)(33,36,39,46)(34,41,38,45)(35,43,37,44)(40,47,42,48);(1,3)(2,8)(4,13)(5,11)(6,9)(7,14)(10,12)(15,16)(17,21)(18,26)(19,20)(22,32)(23,31)(24,29)(25,28)(27,30)(33,37,39,35)(34,40,38,42)(36,43,46,44)(41,48,45,47);(1,2)(3,6)(4,7)(5,9)(8,11)(10,14)(12,15)(13,16)(17,22,23,18)(19,28,27,24)(20,29,30,25)(21,26,31,32)(33,38,39,34)(35,42,37,40)(36,45,46,41)(43,48,44,47)

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=h,d^2=gi,e^2=h,f^2=hi,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, 9117484);