#8093709

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	#52451	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4272 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fghi,c^b=ci,d^b=dh,e^c=eghi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000905);