#8096325

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	#45445	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4379 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=h,e^2=g,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, 8003521);