#451270

order	512
nilpotency class	3
isoclinism	isoclinic class of #418488
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

815 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=ef,d^2=ei,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,h^b=hi,f^c=fi,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358466);