#500891

order	512
nilpotency class	4
isoclinism	isoclinic class of #500891
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

757 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=eh,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.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408087);