#9127357

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	#48988	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4016 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)(3,11)(4,9)(5,13)(6,7)(8,12)(10,15)(14,16)(17,18,23,27)(19,24,28,31)(20,25,30,29)(21,22,26,32)(33,34)(35,36)(37,40)(38,46)(39,42)(41,47)(43,48)(44,45);(2,7)(4,14)(8,11)(10,13)(17,19)(18,24)(20,22)(21,29)(23,28)(25,26)(27,31)(30,32)(33,35,39,37)(34,36,42,40)(38,43,47,45)(41,44,46,48);(2,7)(3,12)(9,16)(10,13)(19,28)(20,30)(24,31)(25,29)(33,36,39,40)(34,37,42,35)(38,44,47,48)(41,45,46,43);(1,3)(2,8)(4,10)(5,9)(6,12)(7,11)(13,14)(15,16)(17,20)(18,25)(19,22)(21,31)(23,30)(24,26)(27,29)(28,32)(33,37,39,35)(34,40,42,36)(38,45,47,43)(41,48,46,44);(1,4,6,14)(2,9,7,16)(3,13,12,10)(5,8,15,11)(17,21)(18,22)(19,29)(20,24)(23,26)(25,28)(27,32)(30,31)(33,34)(35,40)(36,37)(38,46)(39,42)(41,47)(43,44)(45,48);(1,5,6,15)(2,10,7,13)(3,9,12,16)(4,11,14,8)(17,22)(18,26)(19,20)(21,27)(23,32)(24,25)(28,30)(29,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,47)(40,48)(42,46)

Polycyclic presentation:

2^9|a^2=ghi,b^2=gh,c^2=gh,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,d^c=dgi,e^d=egi,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, 9034553);