#8093778

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	#45976	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4449 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=gh,d^2=g,e^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, 8000974);