#432462

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

953 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=fi,c^2=f,d^2=efi,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,h^d=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, 339658);