#438331

order	512
nilpotency class	3
isoclinism	isoclinic class of #422131
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1171 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)(7,13)(8,10)(9,23)(11,15)(12,16)(14,25)(17,21)(18,32)(19,22)(20,28)(24,30)(26,29)(27,34)(31,33)(35,36);(1,2)(3,4)(5,7,11,20,12,21,10,19)(6,13,15,28,16,17,8,22)(9,18,25,34,26,35,24,33)(14,27,29,36,30,31,23,32);(1,2,4,3)(5,8,11,16,12,15,10,6)(7,17,20,28,21,13,19,22)(9,23,25,30,26,29,24,14)(18,31,34,36,35,27,33,32);(1,3,4,2)(5,9)(6,14)(7,18)(8,23)(10,24)(11,25)(12,26)(13,27)(15,29)(16,30)(17,31)(19,33)(20,34)(21,35)(22,32)(28,36)

Polycyclic presentation:

2^9|b^2=eg,c^2=fi,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^c=gi,h^d=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, 345527);