#8749984

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

4023 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)(3,10,5,8)(4,9,12,14)(11,16,13,15)(17,18,23,27)(19,24,22,26)(20,25,31,21)(28,30,29,32)(33,34,39,43)(35,42,37,40)(36,41,47,38)(44,48,46,45);(1,3,7,5)(2,8,6,10)(4,11,12,13)(9,15,14,16)(17,19,23,22)(18,24,27,26)(20,29,31,28)(21,30,25,32)(33,35,39,37)(34,40,43,42)(36,46,47,44)(38,45,41,48);(4,12)(9,14)(11,13)(15,16)(17,20,23,31)(18,25,27,21)(19,28,22,29)(24,30,26,32)(33,35,39,37)(34,40,43,42)(36,46,47,44)(38,45,41,48);(1,4)(2,9)(3,11)(5,13)(6,14)(7,12)(8,15)(10,16)(17,20)(18,21)(19,29)(22,28)(23,31)(24,30)(25,27)(26,32)(33,36,39,47)(34,41,43,38)(35,44,37,46)(40,45,42,48);(1,5,7,3)(2,8,6,10)(4,13,12,11)(9,15,14,16)(17,21)(18,20)(19,30)(22,32)(23,25)(24,29)(26,28)(27,31)(33,37,39,35)(34,42,43,40)(36,46,47,44)(38,45,41,48);(1,6,7,2)(3,10,5,8)(4,9,12,14)(11,15,13,16)(17,22,23,19)(18,26,27,24)(20,29,31,28)(21,30,25,32)(33,38,39,41)(34,36,43,47)(35,45,37,48)(40,46,42,44)

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=hi,d^2=h,e^2=g,f^2=gi,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, 8657180);