#9478138

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4303 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9385334);