#8864552

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

3958 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=gh,d^2=i,e^2=gi,f^2=h,b^a=bg,c^a=ch,d^a=dg,e^a=egi,f^a=fh,c^b=ci,e^b=eh,f^b=fgi,d^c=dgi,e^d=egi

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8771748);