#7732466

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

4586 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)(11,12)(13,17)(14,15)(16,20)(18,19)(21,22)(23,24)(25,26)(27,33)(28,31)(29,34)(30,32)(35,40)(36,38)(37,39);(1,3,5,8)(2,6,4,7)(25,27,30,28)(26,31,32,33)(29,36,37,35)(34,40,39,38);(10,15)(11,16)(17,22)(18,23)(26,32)(27,28)(34,39)(35,36);(1,4,5,2)(3,7,8,6)(9,11,14,16)(10,12,15,20)(13,18,21,23)(17,19,22,24)(25,26)(27,31)(28,33)(29,34)(30,32)(35,38)(36,40)(37,39);(9,12)(10,16)(11,15)(13,19)(14,20)(17,23)(18,22)(21,24)(25,28,30,27)(26,33,32,31)(29,36,37,35)(34,40,39,38);(9,13,14,21)(10,17,15,22)(11,18,16,23)(12,19,20,24)(25,29)(26,34)(27,35)(28,36)(30,37)(31,38)(32,39)(33,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=g,d^2=gh,e^2=i,f^2=hi,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7639662);