#7719035

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	C₂⁴	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4077 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=h,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, 7626231);