#9017371

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	#44131	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) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8924567);