#6373713

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	#34641	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3162 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=g,d^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ci,d^b=di

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6280909);