#8097116

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

4155 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)(3,10)(4,8)(5,7)(6,9)(11,16)(12,15)(13,14)(17,18)(19,27)(20,26)(21,25)(22,28)(23,29)(24,30)(31,32)(33,34,39,42)(35,36,37,40)(38,41,46,48)(43,44,45,47);(4,12)(8,15)(11,13)(14,16)(17,19,24,21)(18,25,30,27)(20,29,31,22)(23,26,28,32)(33,35,39,37)(34,40,42,36)(38,43,46,45)(41,47,48,44);(1,3,6,5)(2,7,9,10)(4,11,12,13)(8,14,15,16)(17,20,24,31)(18,26,30,32)(19,29,21,22)(23,25,28,27)(33,34)(35,36)(37,40)(38,41)(39,42)(43,44)(45,47)(46,48);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21,24,19)(18,27,30,25)(20,22,31,29)(23,32,28,26)(33,36,39,40)(34,35,42,37)(38,44,46,47)(41,43,48,45);(1,5,6,3)(2,7,9,10)(4,13,12,11)(8,14,15,16)(17,22)(18,28)(19,20)(21,31)(23,30)(24,29)(25,32)(26,27)(33,37,39,35)(34,40,42,36)(38,43,46,45)(41,44,48,47);(1,2)(3,7)(4,8)(5,10)(6,9)(11,14)(12,15)(13,16)(17,23,24,28)(18,29,30,22)(19,26,21,32)(20,25,31,27)(33,38,39,46)(34,41,42,48)(35,43,37,45)(36,44,40,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004312);