#466684

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
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

830 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)(5,6,8,9)(7,10,11,12)(13,14)(15,20)(16,31)(17,21)(18,23)(19,24)(22,33)(25,26)(27,42)(28,39)(29,36)(30,37)(32,44)(34,40)(35,43)(38,41);(6,9)(10,12)(13,15,26,41,18,29,19,30)(14,20,25,38,23,36,24,37)(16,32,34,27,17,28,33,43)(21,39,22,35,31,44,40,42);(1,2,4,3)(5,7)(6,10)(8,11)(9,12)(13,16,18,17)(14,21,23,31)(15,27,29,43)(19,34,26,33)(20,35,36,42)(22,25,40,24)(28,41,32,30)(37,44,38,39);(1,3,4,2)(7,11)(10,12)(13,17,18,16)(14,22,23,40)(15,28,29,32)(19,34,26,33)(20,35,36,42)(21,25,31,24)(27,41,43,30)(37,39,38,44)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 373880);