#8097825

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	#36677	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4240 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=g,e^2=hi,f^2=hi,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, 8005021);