#8630807

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

4166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8538003);