#8533328

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	#52316	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4230 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=hi,c^2=ghi,d^2=g,e^2=g,f^2=gh,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, 8440524);