#8535375

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

4272 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)(3,4)(9,14)(11,12)(17,18)(19,24)(20,29)(21,25)(22,28)(23,27)(26,32)(30,31)(34,40)(36,46)(41,43)(44,48);(1,2)(3,8)(4,10)(5,9)(6,7)(11,15)(12,16)(13,14)(17,19,23,30)(18,24,27,31)(20,26,21,28)(22,29,32,25)(33,34,39,40)(35,41,38,43)(36,42,46,37)(44,45,48,47);(1,3,6,4)(2,8,7,10)(5,11,13,12)(9,15,14,16)(17,20,23,21)(18,25,27,29)(19,28,30,26)(22,24,32,31)(33,35,39,38)(34,41,40,43)(36,44,46,48)(37,47,42,45);(1,4,6,3)(2,8,7,10)(5,11,13,12)(9,16,14,15)(17,21,23,20)(18,25,27,29)(19,26,30,28)(22,24,32,31)(33,36)(34,42)(35,44)(37,40)(38,48)(39,46)(41,45)(43,47);(1,2)(3,10)(4,8)(5,9)(6,7)(11,16)(12,15)(13,14)(17,22)(18,26)(19,29)(20,31)(21,24)(23,32)(25,30)(27,28)(33,37,39,42)(34,36,40,46)(35,45,38,47)(41,48,43,44);(1,5,6,13)(2,9,7,14)(3,11,4,12)(8,15,10,16)(17,18)(19,24)(20,25)(21,29)(22,26)(23,27)(28,32)(30,31)(33,38,39,35)(34,43,40,41)(36,44,46,48)(37,47,42,45)

Polycyclic presentation:

2^9|b^2=gh,c^2=gi,d^2=h,e^2=ghi,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^c=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, 8442571);