#8898546

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

4143 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,c^2=gi,d^2=gh,f^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, 8805742);