#8508981

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

4154 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)(4,7)(5,8)(6,9)(11,16)(12,14)(13,15)(17,18)(19,25)(20,30)(21,29)(22,27)(23,24)(26,31)(28,32)(37,45)(41,46)(43,44)(47,48);(3,4)(5,13)(7,10)(8,15)(18,24)(21,31)(25,30)(27,28)(33,34,39,38)(35,42,36,40)(37,46,45,41)(43,47,44,48);(1,3,6,4)(2,7,9,10)(5,11,13,12)(8,14,15,16)(17,19,23,20)(18,25,24,30)(21,28,31,27)(22,29,32,26)(33,35,39,36)(34,40,38,42)(37,43,45,44)(41,47,46,48);(1,4,6,3)(2,7,9,10)(5,12,13,11)(8,14,15,16)(17,20,23,19)(18,25,24,30)(21,27,31,28)(22,29,32,26)(33,36,39,35)(34,40,38,42)(37,44,45,43)(41,47,46,48);(1,5,6,13)(2,8,9,15)(3,11,4,12)(7,14,10,16)(17,21)(18,26)(19,28)(20,27)(22,25)(23,31)(24,29)(30,32)(33,37,39,45)(34,41,38,46)(35,43,36,44)(40,47,42,48);(1,2)(3,7)(4,10)(5,8)(6,9)(11,14)(12,16)(13,15)(17,22)(18,27)(19,29)(20,26)(21,25)(23,32)(24,28)(30,31)(33,38,39,34)(35,42,36,40)(37,46,45,41)(43,48,44,47)

Polycyclic presentation:

2^9|b^2=hi,c^2=gi,d^2=gi,e^2=i,f^2=hi,b^a=bg,c^a=ch,d^a=dg,e^a=eghi,c^b=ci,d^b=dh,e^b=egi,f^d=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, 8416177);