#7857923

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

4528 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ghi,d^2=gh,e^2=gh,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,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, 7765119);