#7719325

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
prime radical	#2424	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4334 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,d^2=i,e^2=hi,f^2=g,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,e^b=ei,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, 7626521);