#451624

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

786 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)(9,10,14,19,15,20,13,16)(11,21,23,33,24,17,29,30)(12,25,26,34,27,31,28,18)(22,36,39,32,38,40,37,35);(1,3)(2,5)(4,7)(6,8)(10,16,20,19)(12,26,27,28)(17,30,21,33)(18,34)(22,37,38,39)(25,31)(32,35)(36,40);(3,7)(5,8)(9,11,13,23,15,24,14,29)(10,17,16,30,20,21,19,33)(12,22,28,39,27,38,26,37)(18,32,31,40,34,35,25,36);(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,31)(17,32)(19,25)(20,34)(21,35)(23,39)(24,38)(29,37)(30,40)(33,36)

Polycyclic presentation:

2^9|a^2=fgh,b^2=f,c^2=ef,e^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^c=hi,e^d=ei,f^d=fi,h^d=hi

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, 358820);