#9224361

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	#50757	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4104 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=gh,d^2=i,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, 9131557);