#416147

order	512
nilpotency class	3
isoclinism	isoclinic class of #394026
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

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

1308 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=i,d^2=h,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, 323343);