#7718386

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	#37129	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4688 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)(9,10)(11,17)(12,20)(13,14)(15,18)(16,21)(19,22)(23,24)(26,31)(29,39)(32,38)(33,35);(1,2)(3,6)(4,5)(7,8)(9,10)(11,17)(12,16)(13,14)(15,18)(19,23)(20,21)(22,24)(25,26,30,31)(27,32,36,38)(28,35,37,33)(29,34,39,40);(10,14)(11,18)(16,20)(19,24)(26,31)(28,37)(32,38)(34,40);(1,3,4,7)(2,6,5,8)(9,10)(11,15)(12,16)(13,14)(17,18)(19,22)(20,21)(23,24)(25,27,30,36)(26,32,31,38)(28,34,37,40)(29,35,39,33);(9,11,13,18)(10,15,14,17)(12,19,21,24)(16,22,20,23)(25,28,30,37)(26,33,31,35)(27,34,36,40)(29,38,39,32);(9,12)(10,16)(11,19)(13,21)(14,20)(15,22)(17,23)(18,24)(25,29)(26,34)(27,35)(28,38)(30,39)(31,40)(32,37)(33,36)

Polycyclic presentation:

2^9|b^2=hi,d^2=g,e^2=i,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, 7625582);