#8726318

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, prime radical, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3813 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=g,d^2=g,e^2=ghi,f^2=gi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,d^c=dgi,f^d=fgi,f^e=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, 8633514);