#8093490

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

3979 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)(3,8)(4,9)(5,12)(6,7)(10,15)(11,14)(13,16)(17,18)(19,22)(20,23)(21,25)(24,26)(27,30)(28,31)(29,32)(34,35)(36,44)(41,42)(45,46);(3,11)(4,13)(8,14)(9,16)(17,19,23,26)(18,24,20,22)(21,29,31,27)(25,30,28,32)(33,34,39,35)(36,43,44,40)(37,41,38,42)(45,48,46,47);(2,7)(3,11)(9,16)(10,12)(18,20)(19,26)(25,28)(27,29)(33,35,39,34)(36,43,44,40)(37,42,38,41)(45,48,46,47);(1,3)(2,8)(4,12)(5,9)(6,11)(7,14)(10,13)(15,16)(17,20,23,18)(19,24,26,22)(21,28,31,25)(27,32,29,30)(33,36)(34,40)(35,43)(37,45)(38,46)(39,44)(41,47)(42,48);(1,4,6,13)(2,9,7,16)(3,12,11,10)(5,14,15,8)(17,21,23,31)(18,25,20,28)(19,27,26,29)(22,30,24,32)(33,37,39,38)(34,41,35,42)(36,45,44,46)(40,47,43,48);(1,5)(2,10)(3,9)(4,14)(6,15)(7,12)(8,13)(11,16)(17,22,23,24)(18,26,20,19)(21,30,31,32)(25,29,28,27)(33,38,39,37)(34,42,35,41)(36,46,44,45)(40,48,43,47)

Polycyclic presentation:

2^9|b^2=g,c^2=gi,d^2=i,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^b=eghi,d^c=dghi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000686);