#8505650

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

3945 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=hi,d^2=i,e^2=gh,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=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, 8412846);