#8739322

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

4081 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)(3,5)(4,8)(6,9)(7,11)(10,13)(12,16)(14,15)(17,18,22,27)(19,24,30,23)(20,21,26,25)(28,31,32,29)(35,37)(36,40)(43,45)(44,46);(1,3)(2,7)(4,10)(5,9)(6,11)(8,15)(12,14)(13,16)(17,18)(19,23)(20,25)(21,26)(22,27)(24,30)(28,31)(29,32)(33,34,39,42)(35,36,37,40)(38,41,47,48)(43,44,45,46);(4,12)(8,16)(10,14)(13,15)(17,19,22,30)(18,23,27,24)(20,28,26,32)(21,29,25,31)(33,35,39,37)(34,40,42,36)(38,43,47,45)(41,46,48,44);(1,4,6,12)(2,8,9,16)(3,10,11,14)(5,13,7,15)(17,19)(18,24)(20,28)(21,31)(22,30)(23,27)(25,29)(26,32)(33,36,39,40)(34,35,42,37)(38,46,47,44)(41,45,48,43);(1,5,6,7)(2,3,9,11)(4,13,12,15)(8,10,16,14)(17,20)(18,25)(19,28)(21,27)(22,26)(23,31)(24,29)(30,32)(33,37,39,35)(34,40,42,36)(38,43,47,45)(41,44,48,46);(1,3)(2,5)(4,14)(6,11)(7,9)(8,15)(10,12)(13,16)(17,21,22,25)(18,26,27,20)(19,29,30,31)(23,32,24,28)(33,38,39,47)(34,41,42,48)(35,43,37,45)(36,44,40,46)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 8646518);