#7716043

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

4566 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:

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

Polycyclic presentation:

2^9|b^2=h,d^2=gh,e^2=gh,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, 7623239);