#8505444

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

4172 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=h,c^2=i,d^2=i,e^2=g,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, 8412640);