#8741892

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

4145 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gi,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8649088);