#9224378

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	#49494	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4053 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)(3,10)(4,12)(5,7)(6,9)(8,13)(11,15)(14,16)(17,18,23,25)(19,24,28,32)(20,29,21,22)(26,27,30,31)(33,34)(35,39)(36,42)(37,45)(38,41)(40,46)(43,48)(44,47);(1,3,6,5)(2,7,9,10)(4,11,13,14)(8,15,12,16)(17,19,23,28)(18,24,25,32)(20,30,21,26)(22,27,29,31)(33,35,38,36)(34,39,41,42)(37,43,46,44)(40,47,45,48);(17,20,23,21)(18,22,25,29)(19,26,28,30)(24,31,32,27)(33,36,38,35)(34,39,41,42)(37,44,46,43)(40,47,45,48);(1,4)(2,8)(3,11)(5,14)(6,13)(7,15)(9,12)(10,16)(17,21,23,20)(18,22,25,29)(19,26,28,30)(24,27,32,31);(1,5,6,3)(2,7,9,10)(4,11,13,14)(8,16,12,15)(17,19)(18,24)(20,26)(21,30)(22,31)(23,28)(25,32)(27,29)(33,34)(35,42)(36,39)(37,40)(38,41)(43,48)(44,47)(45,46);(1,2)(3,7)(4,12)(5,10)(6,9)(8,13)(11,16)(14,15)(17,22)(18,21)(19,27)(20,25)(23,29)(24,26)(28,31)(30,32)(33,37)(34,40)(35,43)(36,44)(38,46)(39,47)(41,45)(42,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,c^2=h,d^2=i,e^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, 9131574);