#8726655

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

3986 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)(33,34,39,40)(35,43,38,41)(36,46,42,37)(44,48,47,45);(1,2)(3,8)(4,9)(5,10)(6,7)(11,14)(12,13)(15,16)(17,19,24,30)(18,23,28,32)(20,21,22,29)(25,31,27,26)(34,40)(36,42)(41,43)(44,47);(2,7)(3,9)(10,16)(11,13)(17,20,24,22)(18,25,28,27)(19,29,30,21)(23,26,32,31)(33,35,39,38)(34,41,40,43)(36,47,42,44)(37,45,46,48);(1,3,6,9)(2,8,7,4)(5,13,15,11)(10,12,16,14)(17,21,24,29)(18,26,28,31)(19,20,30,22)(23,27,32,25)(33,36,39,42)(34,37,40,46)(35,44,38,47)(41,48,43,45);(1,4,6,8)(2,9,7,3)(5,14,15,12)(10,11,16,13)(17,22,24,20)(18,27,28,25)(19,29,30,21)(23,26,32,31)(33,37)(34,42)(35,45)(36,40)(38,48)(39,46)(41,44)(43,47);(1,5,6,15)(2,10,7,16)(3,11,9,13)(4,12,8,14)(17,23,24,32)(18,19,28,30)(20,26,22,31)(21,25,29,27)(33,38,39,35)(34,41,40,43)(36,44,42,47)(37,45,46,48)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,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, 8633851);