#501630

order	512
nilpotency class	4
isoclinism	isoclinic class of #500733
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#33589	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

651 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ei,e^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,g^b=gh,h^b=hi,d^c=di,g^c=gh,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, 408826);