#8075494

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

4296 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)(9,10,15,17)(11,18,14,16)(12,13,21,22)(19,23,24,20)(25,26,31,34)(27,32,28,30)(29,33,38,40)(35,39,36,37);(1,2)(3,7)(4,8)(5,6)(9,11,15,14)(10,16,17,18)(12,19,21,24)(13,20,22,23)(25,27,31,28)(26,32,34,30)(29,35,38,36)(33,39,40,37);(10,17)(11,14)(13,22)(19,24)(25,27,31,28)(26,30,34,32)(29,35,38,36)(33,37,40,39);(1,3,5,4)(2,7,6,8)(9,12)(10,13)(11,19)(14,24)(15,21)(16,20)(17,22)(18,23)(25,28,31,27)(26,30,34,32)(29,35,38,36)(33,39,40,37);(1,3,5,4)(2,7,6,8)(9,13)(10,12)(11,20)(14,23)(15,22)(16,19)(17,21)(18,24)(25,29,31,38)(26,33,34,40)(27,35,28,36)(30,37,32,39);(1,4,5,3)(2,8,6,7)(9,14,15,11)(10,16,17,18)(12,19,21,24)(13,23,22,20)(25,30)(26,27)(28,34)(29,37)(31,32)(33,35)(36,40)(38,39)

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=hi,d^2=gh,e^2=gh,f^2=g,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, 7982690);