#8095210

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	#34876	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4312 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=gi,e^2=gi,f^2=h,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, 8002406);