#8535312

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
prime radical	#57748	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3953 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,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, 8442508);