#8726374

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	#2682	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=ghi,c^2=h,d^2=gh,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, 8633570);