#439260

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

883 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)(5,6,9,8)(7,11,12,10)(13,14,17,21)(15,23,26,19)(16,28,30,42)(18,32,24,22)(20,36,37,31)(25,40,29,43)(27,35,33,38)(34,41,39,44);(5,7)(6,10)(8,11)(9,12)(13,15,24,27,17,26,18,33)(14,19,22,38,21,23,32,35)(16,29,31,41,30,25,36,44)(20,34,42,43,37,39,28,40);(5,6,9,8)(7,10,12,11)(13,16,24,36,17,30,18,31)(14,20,32,42,21,37,22,28)(15,25,33,44,26,29,27,41)(19,34,38,40,23,39,35,43);(2,4)(5,8,9,6)(7,10,12,11)(15,26)(19,23)(25,29)(27,33)(34,39)(35,38)(40,43)(41,44)

Polycyclic presentation:

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

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, 346456);