#437899

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
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	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1142 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=gi,c^2=ei,d^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,f^c=fi,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, 345095);