#7916624

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

4636 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)(9,10,13,14)(11,17,19,15)(12,16,21,22)(18,24,20,23)(25,26,31,35)(27,34,36,30)(28,32,39,40)(29,37,38,33);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,27)(26,30)(28,37)(29,32)(31,36)(33,39)(34,35)(38,40);(1,3,5,4)(2,7,6,8)(10,14)(15,17)(16,22)(23,24)(25,28,31,39)(26,32,35,40)(27,33,36,37)(29,30,38,34);(9,10)(11,15)(12,16)(13,14)(17,19)(18,23)(20,24)(21,22)(25,29,31,38)(26,33,35,37)(27,32,36,40)(28,30,39,34);(1,3,5,4)(2,7,6,8)(9,11)(10,15)(12,20)(13,19)(14,17)(16,24)(18,21)(22,23)(25,30)(26,27)(28,38)(29,39)(31,34)(32,33)(35,36)(37,40);(1,4,5,3)(2,8,6,7)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,27)(26,34)(28,33)(29,32)(30,35)(31,36)(37,39)(38,40)

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=hi,e^2=gi,f^2=ghi,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=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, 7823820);