#8726261

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3898 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

(17,18)(19,23)(20,27)(21,26)(22,25)(24,28)(29,31)(30,32)(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)(17,19,24,30)(18,23,28,32)(20,21,22,29)(25,31,27,26)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(2,7)(3,10)(9,16)(11,13)(17,20,24,22)(18,25,28,27)(19,29,30,21)(23,26,32,31)(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)(17,21,24,29)(18,26,28,31)(19,20,30,22)(23,27,32,25)(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)(17,22,24,20)(18,27,28,25)(19,29,30,21)(23,26,32,31)(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)(17,23,24,32)(18,19,28,30)(20,26,22,31)(21,25,29,27)(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=gi,c^2=h,d^2=h,e^2=i,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, 8633457);