#9224321

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	#59738	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4082 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=h,d^2=gi,e^2=g,f^2=g,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, 9131517);