#416282

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
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)(15,25)(16,30)(17,24)(18,23)(19,22)(20,27)(21,33)(26,41)(28,36)(29,37)(31,35)(32,38)(34,39)(40,43)(42,44);(6,9)(10,11)(13,15,18,28,19,29,17,27)(14,20,24,37,22,36,23,25)(16,31,33,42,34,26,32,43)(21,35,30,40,38,41,39,44);(2,4)(13,16,19,34)(14,21,22,38)(15,26,29,31)(17,33,18,32)(20,35,36,41)(23,30,24,39)(25,40,37,44)(27,43,28,42);(5,7,8,12)(6,10,9,11)(14,22)(20,36)(21,38)(23,24)(25,37)(30,39)(35,41)(40,44)

Polycyclic presentation:

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

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, 323478);