#8993228

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

4210 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=gi,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, 8900424);