#7710008

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

4258 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7617204);