#8096119

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

4373 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=g,e^2=i,f^2=i,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, 8003315);