#425285

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1071 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)(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,30)(17,23)(18,22)(19,24)(20,27)(21,33)(26,35)(28,37)(29,36)(31,44)(32,40)(34,39)(38,41)(42,43);(13,15,19,29)(14,20,24,37)(16,31,34,41)(17,28,18,27)(21,38,40,44)(22,36,23,25)(26,33,42,32)(30,43,39,35);(5,7,8,12)(6,9,10,11)(13,14)(15,25)(16,21)(17,22)(18,23)(19,24)(20,28)(26,38)(27,37)(29,36)(30,33)(31,43)(32,39)(34,40)(35,41)(42,44);(2,4)(13,16,18,33,19,34,17,32)(14,21,23,30,24,40,22,39)(15,26,28,41,29,42,27,31)(20,35,36,44,37,43,25,38)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=f,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=gi,h^b=hi,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, 332481);