#501624

order	512
nilpotency class	4
isoclinism	isoclinic class of #501014
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
prime radical	C₂×#509	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×C₄⋊C₄ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

661 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=i,d^2=ei,f^2=i,g^2=hi,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, 408820);