#8096541

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	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4271 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=g,d^2=h,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, 8003737);