#8097247

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

4157 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=h,d^2=g,e^2=gh,f^2=h,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, 8004443);