#453916

order	512
nilpotency class	3
isoclinism	isoclinic class of #420064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2495
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

545 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)(3,5)(4,6)(7,8)(9,10,15,19,16,20,14,13)(11,21,25,17,26,31,24,23)(12,27,29,35,30,36,32,18)(22,33,28,34,39,37,38,40);(1,3,4,7)(2,5,6,8)(9,11,15,24,16,26,14,25)(10,17,19,21,20,23,13,31)(12,28,32,39,30,38,29,22)(18,34,36,37,35,40,27,33);(9,12,15,32,16,30,14,29)(10,18,19,36,20,35,13,27)(11,22,24,28,26,39,25,38)(17,33,21,34,23,37,31,40);(2,6)(5,8)(9,13,14,20,16,19,15,10)(11,23,24,31,26,17,25,21)(12,27,29,35,30,36,32,18)(22,37,28,40,39,33,38,34)

Polycyclic presentation:

2^9|a^2=g,b^2=efgh,c^2=efg,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^b=gi,f^c=fi,g^c=gi

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, 361112);