#438761

order	512
nilpotency class	3
isoclinism	isoclinic class of #417784
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

878 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)(5,6,9,8)(7,11,12,10)(13,14,19,24)(15,20,29,37)(16,21,31,40)(17,23,18,33)(22,32,39,30)(25,34,41,35)(26,38,42,44)(27,36,28,43);(1,2)(3,4)(5,7,9,12)(6,10,8,11)(13,15,17,27,19,29,18,28)(14,20,23,36,24,37,33,43)(16,26,30,41,31,42,32,25)(21,38,22,35,40,44,39,34);(1,2,4,3)(13,16)(14,21)(15,25)(17,32)(18,30)(19,31)(20,34)(22,33)(23,39)(24,40)(26,28)(27,42)(29,41)(35,37)(36,44)(38,43);(1,3,4,2)(5,8,9,6)(7,10,12,11)(13,16)(14,22)(15,26)(17,30)(18,32)(19,31)(20,35)(21,33)(23,40)(24,39)(25,28)(27,41)(29,42)(34,37)(36,44)(38,43)

Polycyclic presentation:

2^9|a^2=efi,b^2=eg,c^2=f,d^2=e,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, 345957);