#8094892

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	#45641	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4189 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=h,e^2=hi,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, 8002088);