#8726641

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	#2407	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3973 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:

(19,21)(24,26)(27,29)(31,32)(33,34,38,41)(35,42,37,39)(36,44,46,40)(43,47,45,48);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,13)(15,16)(17,18)(19,26)(20,23)(21,24)(22,25)(27,32)(28,30)(29,31)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(2,7)(3,9)(10,16)(11,13)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32)(33,35,38,37)(34,39,41,42)(36,45,46,43)(40,48,44,47);(1,3,6,9)(2,8,7,4)(5,13,15,11)(10,12,16,14)(17,20,23,18)(19,24,21,26)(22,25,30,28)(27,32,29,31)(33,36,38,46)(34,40,41,44)(35,43,37,45)(39,47,42,48);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,16,13)(17,21,23,19)(18,24,20,26)(22,27,30,29)(25,32,28,31)(33,34)(35,39)(36,40)(37,42)(38,41)(43,47)(44,46)(45,48);(1,5,6,15)(2,10,7,16)(3,11,9,13)(4,12,8,14)(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)(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=g,c^2=h,d^2=gi,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, 8633837);