#7771952

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

4545 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)(14,16)(15,19)(18,20)(21,23)(22,24)(25,26)(27,28)(29,33)(30,31)(32,36)(34,35)(37,38)(39,40);(1,3,5,8)(2,6,4,7)(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,26,30,31)(27,28,32,36)(29,33,37,38)(34,35,39,40);(1,4,5,2)(3,7,8,6)(10,16)(12,15)(17,23)(20,22)(26,31)(27,32)(29,37)(35,40);(9,12,14,15)(10,11,16,19)(13,20,21,22)(17,18,23,24)(25,27)(26,28)(29,34)(30,32)(31,36)(33,35)(37,39)(38,40);(9,11)(10,15)(12,16)(13,18)(14,19)(17,22)(20,23)(21,24)(25,28)(26,32)(27,31)(29,35)(30,36)(33,39)(34,38)(37,40);(9,13,14,21)(10,17,16,23)(11,18,19,24)(12,20,15,22)(25,29,30,37)(26,33,31,38)(27,34,32,39)(28,35,36,40)

Polycyclic presentation:

2^9|a^2=gh,b^2=ghi,c^2=gh,d^2=hi,f^2=h,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^c=dh,f^c=fi,e^d=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, 7679148);