#9232832

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	#43185	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4098 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,d^2=i,e^2=gh,f^2=gi,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,e^d=egi,f^d=fgi,f^e=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9140028);