#7973026

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

4486 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=hi,d^2=g,f^2=h,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7880222);