#438695

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

857 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:

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

Polycyclic presentation:

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

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, 345891);