#7743631

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

4515 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)(10,11)(15,17)(16,18)(22,23)(25,26,32,31)(27,37,39,33)(28,35,29,34)(30,40,38,36);(1,2,5,6)(3,7,4,8)(9,10,14,11)(12,15,19,17)(13,18,21,16)(20,23,24,22)(25,27)(26,33)(28,30)(29,38)(31,37)(32,39)(34,40)(35,36);(10,11)(12,19)(16,18)(20,24)(25,28,32,29)(26,34,31,35)(27,38,39,30)(33,36,37,40);(1,3,5,4)(2,7,6,8)(9,11,14,10)(12,17,19,15)(13,18,21,16)(20,23,24,22)(25,29,32,28)(26,35,31,34)(27,38,39,30)(33,36,37,40);(1,4,5,3)(2,8,6,7)(9,12,14,19)(10,15,11,17)(13,20,21,24)(16,22,18,23)(25,30,32,38)(26,36,31,40)(27,28,39,29)(33,35,37,34);(9,13,14,21)(10,16,11,18)(12,20,19,24)(15,22,17,23)(25,31,32,26)(27,33,39,37)(28,35,29,34)(30,40,38,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7650827);