#8734507

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

3970 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)(3,7)(4,13)(5,9)(6,10)(8,12)(11,16)(14,15)(17,18)(19,29)(20,26)(21,22)(23,28)(24,31)(25,32)(27,30)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,3,6,11)(2,7,10,16)(4,5,12,15)(8,14,13,9)(17,19)(18,24)(20,21)(22,25)(23,31)(26,27)(28,29)(30,32)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(3,11)(4,12)(7,16)(8,13)(17,20,23,32)(18,25,28,26)(19,21,31,30)(22,29,27,24)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,4)(2,8)(3,5)(6,12)(7,14)(9,16)(10,13)(11,15)(17,20)(18,26)(19,30)(21,31)(22,24)(23,32)(25,28)(27,29)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,2)(3,7)(4,8)(5,14)(6,10)(9,15)(11,16)(12,13)(17,21,23,30)(18,27,28,22)(19,20,31,32)(24,26,29,25)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,5)(2,9)(3,12)(4,11)(6,15)(7,8)(10,14)(13,16)(17,22,23,27)(18,21,28,30)(19,25,31,26)(20,29,32,24)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)

Polycyclic presentation:

2^9|a^2=gh,b^2=i,c^2=h,d^2=gh,e^2=g,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, 8641703);