#505732

order	512
nilpotency class	4
isoclinism	isoclinic class of #500891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3050	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

541 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)(10,17)(11,23,25,40)(12,26)(13,15)(14,28)(18,32,35,37)(19,27)(21,22)(24,29,30,39)(31,34,38,33);(1,3)(2,5)(4,7)(6,8)(9,10,13,17,16,22,15,21)(11,18,24,31,25,35,30,38)(12,27,26,36,28,19,14,20)(23,34,39,32,40,33,29,37);(3,7)(5,8)(9,11)(10,18)(12,23)(13,24)(14,29)(15,30)(16,25)(17,31)(19,33)(20,37)(21,38)(22,35)(26,39)(27,34)(28,40)(32,36);(9,12,13,26,16,28,15,14)(10,19,17,20,22,27,21,36)(11,23,24,39,25,40,30,29)(18,33,31,37,35,34,38,32)

Polycyclic presentation:

2^9|a^2=egh,b^2=e,d^2=e,e^2=i,f^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,f^a=fh,h^a=hi,c^b=cg,d^b=di,f^c=fhi,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72265609, #72265615, #72265621
maximal normal subgroup of	#72265609, #72265615, #72265621

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 412928);