#8075421

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

4537 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ghi,d^2=h,e^2=h,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, 7982617);