#7783950

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

4474 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=h,c^2=h,d^2=gi,e^2=gh,f^2=gi,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, 7691146);