#7927093

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

4418 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)(9,10,14,12)(11,17,19,15)(13,16,21,20)(18,23,24,22)(25,26,30,32)(27,31,33,28)(29,34,38,39)(35,37,40,36);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,27,30,33)(26,31,32,28)(29,35,38,40)(34,37,39,36);(10,12)(15,17)(16,20)(22,23)(26,32)(27,33)(34,39)(35,40);(1,2,5,4)(3,6,8,7)(9,12,14,10)(11,17,19,15)(13,20,21,16)(18,23,24,22)(25,26)(27,31)(28,33)(29,34)(30,32)(35,37)(36,40)(38,39);(9,11)(10,15)(12,17)(13,18)(14,19)(16,22)(20,23)(21,24)(25,28,30,31)(26,33,32,27)(29,37,38,36)(34,35,39,40);(1,4,5,2)(3,7,8,6)(9,13,14,21)(10,16,12,20)(11,18,19,24)(15,22,17,23)(25,29,30,38)(26,34,32,39)(27,35,33,40)(28,36,31,37)

Polycyclic presentation:

2^9|a^2=gi,b^2=gh,d^2=g,e^2=i,f^2=gi,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, 7834289);