#7720211

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

4399 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)(9,10,15,12)(11,19,20,16)(13,17,22,23)(14,24,21,18)(25,26,30,34)(27,32,37,31)(28,38,33,29)(35,40,39,36);(1,3,5,8)(2,6,4,7)(9,11,15,20)(10,16,12,19)(13,14,22,21)(17,18,23,24)(25,27)(26,31)(28,39)(29,36)(30,37)(32,34)(33,35)(38,40);(11,20)(13,22)(16,19)(17,23)(25,27,30,37)(26,31,34,32)(28,39,33,35)(29,36,38,40);(1,2,5,4)(3,6,8,7)(9,12,15,10)(11,19,20,16)(13,23,22,17)(14,24,21,18)(25,27)(26,32)(28,35)(29,36)(30,37)(31,34)(33,39)(38,40);(1,4,5,2)(3,7,8,6)(9,13,15,22)(10,17,12,23)(11,21,20,14)(16,24,19,18)(25,28,30,33)(26,29,34,38)(27,35,37,39)(31,40,32,36);(9,14,15,21)(10,18,12,24)(11,22,20,13)(16,23,19,17)(25,29)(26,33)(27,36)(28,34)(30,38)(31,35)(32,39)(37,40)

Polycyclic presentation:

2^9|a^2=g,b^2=gh,c^2=h,d^2=gh,e^2=g,f^2=hi,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, 7627407);