#8147847

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

4233 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)(3,8)(5,13)(11,16)(17,18)(19,22)(20,25)(21,24)(23,28)(26,29)(27,31)(30,32)(38,46)(41,45)(43,44)(47,48);(1,2,6,7)(3,4,8,12)(5,9,13,14)(10,11,15,16)(18,19)(23,25)(24,26)(30,31)(33,34,39,37)(35,42,36,40)(38,41,46,45)(43,48,44,47);(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,30,32,31)(33,35,39,36)(34,40,37,42)(38,43,46,44)(41,47,45,48);(1,3)(2,4)(5,10)(6,8)(7,12)(9,11)(13,15)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30)(33,36,39,35)(34,40,37,42)(38,44,46,43)(41,47,45,48);(1,4)(2,8)(3,7)(5,11)(6,12)(9,15)(10,14)(13,16)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31)(33,37,39,34)(35,42,36,40)(38,45,46,41)(43,48,44,47);(1,5)(2,9)(3,10)(4,11)(6,13)(7,14)(8,15)(12,16)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)(33,38)(34,41)(35,43)(36,44)(37,45)(39,46)(40,47)(42,48)

Polycyclic presentation:

2^9|b^2=gi,c^2=i,d^2=i,e^2=i,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=eh,f^a=fgi,c^b=ci,d^b=dh,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8055043);