#430826

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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

874 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)(13,14,18,23)(15,26,29,20)(16,32,17,21)(19,24,36,25)(22,34,40,33)(27,37,43,41)(28,39,35,44)(30,42,31,38);(5,7,9,12)(6,10,8,11)(13,15,18,29)(14,20,23,26)(16,28,17,35)(19,31,36,30)(21,39,32,44)(22,41,40,37)(24,42,25,38)(27,34,43,33);(1,2,4,3)(5,8,9,6)(7,11,12,10)(13,16,18,17)(14,21,23,32)(15,27,29,43)(19,34,36,33)(20,37,26,41)(22,25,40,24)(28,30,35,31)(38,44,42,39);(1,3,4,2)(7,12)(10,11)(13,17,18,16)(14,22,23,40)(15,28,29,35)(19,34,36,33)(20,37,26,41)(21,25,32,24)(27,30,43,31)(38,39,42,44)

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=efi,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^b=de,g^b=gi,g^c=gi,h^d=hi

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, 338022);