#453772

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

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

614 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:

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

Polycyclic presentation:

2^9|a^2=efgh,b^2=fg,c^2=efg,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,h^c=hi,e^d=ei,f^d=fi,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, 360968);