#8075556

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4410 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:

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

Polycyclic presentation:

2^9|b^2=g,c^2=ghi,d^2=gh,e^2=g,f^2=gh,b^a=bg,c^a=ch,e^a=ei,c^b=ci,e^c=ei,f^c=fh,e^d=ehi,f^d=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7982752);