#9206500

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	#2464	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(3,12)(4,10)(5,11)(6,8)(7,9)(13,14)(15,16)(17,18,23,28)(19,22,27,24)(20,25,30,32)(21,26,29,31)(33,34)(35,42)(36,44)(37,40)(38,39)(41,46)(43,47)(45,48);(1,3,7,6)(2,8,9,12)(4,11,14,15)(5,10,16,13)(17,19)(18,24)(20,29)(21,30)(22,28)(23,27)(25,26)(31,32)(35,37)(40,42)(43,45)(47,48);(2,9)(4,14)(8,12)(11,15)(17,20,23,30)(18,25,28,32)(19,29,27,21)(22,31,24,26)(33,35,39,37)(34,40,38,42)(36,45,46,43)(41,48,44,47);(1,4,7,14)(2,10,9,13)(3,11,6,15)(5,8,16,12)(17,20)(18,25)(19,29)(21,27)(22,31)(23,30)(24,26)(28,32)(33,36,39,46)(34,41,38,44)(35,43,37,45)(40,47,42,48);(1,5,7,16)(2,11,9,15)(3,13,6,10)(4,12,14,8)(17,21,23,29)(18,26,28,31)(19,30,27,20)(22,32,24,25)(33,37,39,35)(34,40,38,42)(36,43,46,45)(41,48,44,47);(1,6,7,3)(2,12,9,8)(4,15,14,11)(5,13,16,10)(17,22)(18,27)(19,28)(20,31)(21,25)(23,24)(26,30)(29,32)(33,38,39,34)(35,42,37,40)(36,44,46,41)(43,48,45,47)

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,c^2=gh,d^2=h,e^2=gi,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, 9113696);