#8271887

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

4235 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:

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

Polycyclic presentation:

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

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, 8179083);