#8093302

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	#45515	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4396 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=g,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^b=eghi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000498);