#8818026

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	#59806	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4227 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8725222);