#8893816

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

4162 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=h,c^2=g,e^2=gh,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,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, 8801012);