#8581807

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

4099 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)(3,4)(5,9)(6,7)(8,10)(11,12)(13,16)(14,15)(17,18,23,20)(19,21,24,26)(22,25,30,28)(27,29,31,32)(34,38)(35,41)(40,43)(42,47);(1,3)(2,4)(5,11)(6,10)(7,8)(9,12)(13,15)(14,16)(17,19)(18,21)(20,26)(22,27)(23,24)(25,29)(28,32)(30,31)(33,34,37,38)(35,39,41,45)(36,40,44,43)(42,46,47,48);(2,7)(3,10)(9,15)(11,16)(17,18,23,20)(19,26,24,21)(22,25,30,28)(27,32,31,29)(33,35,37,41)(34,39,38,45)(36,42,44,47)(40,46,43,48);(1,4,6,8)(2,3,7,10)(5,13,14,12)(9,16,15,11)(17,20,23,18)(19,26,24,21)(22,25,30,28)(27,29,31,32)(33,34)(35,39)(36,43)(37,38)(40,44)(41,45)(42,48)(46,47);(1,3)(2,8)(4,7)(5,11)(6,10)(9,13)(12,15)(14,16)(17,21,23,26)(18,24,20,19)(22,29,30,32)(25,31,28,27)(33,35)(34,39)(36,42)(37,41)(38,45)(40,46)(43,48)(44,47);(1,5)(2,9)(3,11)(4,12)(6,14)(7,15)(8,13)(10,16)(17,22,23,30)(18,25,20,28)(19,27,24,31)(21,29,26,32)(33,36,37,44)(34,40,38,43)(35,42,41,47)(39,46,45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=hi,d^2=i,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^c=egi,f^d=fgi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8489003);