#503982

order	512
nilpotency class	4
isoclinism	isoclinic class of #503866
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#481
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#28487	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#436 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

600 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:

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

Polycyclic presentation:

2^9|b^2=fh,c^2=f,d^2=ei,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=de,g^a=gi,c^b=cg,f^b=fi,g^b=gh,h^b=hi,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, 411178);