#8889465

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

4121 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)(3,9)(4,12)(5,6)(7,10)(8,14)(11,15)(13,16)(17,18,22,23)(19,24,25,20)(21,29,30,26)(27,28,32,31)(34,38)(35,41)(36,43)(47,48);(2,6)(7,9)(8,12)(15,16)(17,18,22,23)(19,24,25,20)(21,29,30,26)(27,28,32,31)(33,34,37,38)(35,39,41,46)(36,44,43,40)(42,48,45,47);(3,10)(7,9)(11,13)(15,16)(18,23)(19,25)(26,29)(27,32)(33,35,37,41)(34,39,38,46)(36,42,43,45)(40,47,44,48);(1,3)(2,7)(4,13)(5,10)(6,9)(8,16)(11,14)(12,15)(17,19)(18,24)(20,23)(21,27)(22,25)(26,31)(28,29)(30,32)(33,35)(34,39)(36,45)(37,41)(38,46)(40,48)(42,43)(44,47);(1,4)(2,8)(3,11)(5,14)(6,12)(7,15)(9,16)(10,13)(17,20,22,24)(18,25,23,19)(21,28,30,31)(26,32,29,27)(33,36)(34,40)(35,42)(37,43)(38,44)(39,47)(41,45)(46,48);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,14)(13,16)(17,21,22,30)(18,26,23,29)(19,27,25,32)(20,28,24,31)(33,34)(35,39)(36,40)(37,38)(41,46)(42,47)(43,44)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8796661);