#501628

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
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1772 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

671 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=e,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, 408824);