#7743826

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	#2431	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4270 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,d^2=g,e^2=h,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, 7651022);