#9021961

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3829 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=hi,c^2=h,d^2=g,e^2=g,f^2=i,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, 8929157);