#8893421

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

4285 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=g,c^2=gh,e^2=g,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, 8800617);