#8014745

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	#45994	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4307 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,c^2=i,d^2=g,e^2=ghi,f^2=g,b^a=bg,c^a=ch,f^a=fh,c^b=ci,e^b=eh,d^c=dh,f^c=fi,e^d=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7921941);