#453422

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

638 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:

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

Polycyclic presentation:

2^9|a^2=egh,b^2=ef,c^2=eg,d^2=e,f^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,h^d=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 360618);