#504264

order	512
nilpotency class	4
isoclinism	isoclinic class of #504125
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#476
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

532 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)(5,6,11,13,8,14,9,15)(7,17,26,28,19,12,20,30)(10,16,25,31,22,32,23,33)(18,34,35,29,24,36,21,27);(1,2,4,3)(5,7,9,20,8,19,11,26)(6,12,15,28,14,17,13,30)(10,24,23,35,22,18,25,21)(16,34,33,27,32,36,31,29);(1,2,4,3)(6,13)(7,18,19,24)(9,11)(10,25)(12,27,17,29)(14,15)(16,32)(20,35,26,21)(22,23)(28,34,30,36);(1,3,4,2)(6,14)(12,17)(13,15)(16,32)(27,29)(28,30)(31,33)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 411460);