#8097818

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4708 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

83 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8005014);