#451331

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

806 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)(9,10,15,20,14,19,13,17)(11,21,23,32,24,16,28,30)(12,25,26,35,27,18,29,33)(22,34,38,39,37,31,36,40);(1,3)(2,5)(4,7)(6,8)(9,11,13,23,14,24,15,28)(10,16,17,30,19,21,20,32)(12,22,26,36,27,37,29,38)(18,34,33,40,25,31,35,39);(3,7)(5,8)(10,17,19,20)(16,30,21,32)(18,33,25,35)(31,39,34,40);(9,12)(10,18,19,25)(11,22)(13,26)(14,27)(15,29)(16,31,21,34)(17,33,20,35)(23,36)(24,37)(28,38)(30,39,32,40)

Polycyclic presentation:

2^9|a^2=fgh,b^2=ef,c^2=e,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, 358527);