#8656204

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

4188 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=g,d^2=hi,e^2=hi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,e^c=egi,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8563400);