#8097255

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	#45556	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4409 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8004451);