#506412

order	512
nilpotency class	4
isoclinism	isoclinic class of #500681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#459
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

724 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)(8,23)(9,11)(10,25)(14,29)(15,24)(17,18)(20,22)(21,35)(26,27)(28,30)(31,36)(33,34);(1,2,3,4)(5,6,9,13,12,18,11,17)(7,14,21,28,22,31,27,33)(8,24,23,32,25,15,10,16)(19,30,26,29,20,34,35,36);(2,4)(5,7)(6,14)(8,20)(9,21)(10,26)(11,27)(12,22)(13,28)(15,30)(16,29)(17,33)(18,31)(19,25)(23,35)(24,34)(32,36);(5,8,11,10,12,25,9,23)(6,15,17,32,18,24,13,16)(7,20,27,26,22,19,21,35)(14,30,33,36,31,34,28,29)

Polycyclic presentation:

2^9|b^2=eg,d^2=ei,e^2=i,f^2=hi,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=fh,h^c=hi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72238786, #72239803, #72239829
maximal normal subgroup of	#72238786, #72239803, #72239829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 413608);