#501650

order	512
nilpotency class	4
isoclinism	isoclinic class of #501052
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#466
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2221	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

655 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)(5,6,9,14,22,32,35,30,8,13,21,31,10,15,23,12)(7,16,19,25,20,28,34,29,18,11,24,26,17,27,33,36);(1,2,4,3)(5,7,8,18)(6,11,13,16)(9,19,21,24)(10,17,22,20)(12,29,30,36)(14,26,31,25)(15,28,32,27)(23,33,35,34);(6,12)(7,17)(9,23)(10,22)(11,25)(13,30)(14,15)(16,26)(18,20)(19,24)(21,35)(27,36)(28,29)(31,32);(1,3,4,2)(6,13)(11,16)(12,30)(14,31)(15,32)(25,26)(27,28)(29,36)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 408846);