#416253

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

944 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,9)(7,11,12,10)(13,14,17,23,19,22,18,24)(15,20,26,36,27,35,33,40)(16,28,30,38,32,39,31,21)(25,37,29,43,41,44,42,34);(1,3,4,2)(6,9)(10,11)(13,15,19,27)(14,20,22,35)(16,29,32,42)(17,26,18,33)(21,37,38,44)(23,36,24,40)(25,30,41,31)(28,43,39,34);(2,3)(13,16,19,32)(14,21,22,38)(15,25,27,41)(17,31,18,30)(20,34,35,43)(23,39,24,28)(26,42,33,29)(36,44,40,37);(5,7,8,12)(6,10,9,11)(14,22)(20,35)(21,38)(23,24)(28,39)(34,43)(36,40)(37,44)

Polycyclic presentation:

2^9|a^2=ef,b^2=eh,c^2=i,d^2=hi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323449);