#7732821

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=gi,d^2=g,e^2=hi,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, 7640017);