#7714651

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

4329 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)(9,10,14,15)(11,12,17,20)(13,16,21,22)(18,19,23,24)(25,26)(27,35)(28,30)(29,32)(31,34)(33,40)(36,37)(38,39);(1,2,4,5)(3,6,7,8)(25,27,31,29)(26,32,34,35)(28,38,33,36)(30,37,40,39);(10,15)(11,17)(16,22)(18,23)(27,29)(32,35)(36,38)(37,39);(1,3,4,7)(2,6,5,8)(9,11)(10,12)(13,18)(14,17)(15,20)(16,19)(21,23)(22,24)(25,28,31,33)(26,30,34,40)(27,36,29,38)(32,39,35,37);(9,12)(10,11)(13,19)(14,20)(15,17)(16,18)(21,24)(22,23)(25,29,31,27)(26,32,34,35)(28,38,33,36)(30,39,40,37);(9,13,14,21)(10,16,15,22)(11,18,17,23)(12,19,20,24)(25,30)(26,33)(27,37)(28,34)(29,39)(31,40)(32,36)(35,38)

Polycyclic presentation:

2^9|a^2=h,b^2=g,d^2=g,e^2=i,f^2=h,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^b=di,d^c=dh,e^c=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, 7621847);