#7806593

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

4534 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)(9,10,14,11)(12,15,19,17)(13,16,21,18)(20,22,24,23)(25,26)(27,32)(28,34)(29,30)(31,35)(33,39)(36,40)(37,38);(1,2)(3,6)(4,5)(7,8)(10,11)(12,19)(13,21)(22,23)(25,27,31,39)(26,32,35,33)(28,38,30,36)(29,40,34,37);(1,3,4,7)(2,6,5,8)(9,11,14,10)(12,17,19,15)(13,16,21,18)(20,22,24,23)(25,27)(26,33)(28,38)(29,37)(30,36)(31,39)(32,35)(34,40);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,28,31,30)(26,34,35,29)(27,36,39,38)(32,40,33,37);(9,12)(10,15)(11,17)(13,20)(14,19)(16,22)(18,23)(21,24)(25,29)(26,28)(27,37)(30,35)(31,34)(32,36)(33,38)(39,40);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,30,31,28)(26,29,35,34)(27,38,39,36)(32,37,33,40)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7713789);