#7927092

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

4440 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=gi,f^2=i,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, 7834288);