#7932334

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

4450 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=gh,e^2=gh,f^2=h,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, 7839530);