#460926

order	512
nilpotency class	3
isoclinism	isoclinic class of #457092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

973 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)(7,9)(8,10)(11,12)(13,14,17,20,19,24,18,23)(15,25,27,21,29,35,28,37)(16,30,32,40,34,39,33,22)(26,36,31,44,42,43,41,38);(5,7,8,11)(6,9,10,12)(14,20,24,23)(21,35,37,25)(22,30,40,39)(36,44,43,38);(6,10)(9,12)(13,15,19,29)(14,21,24,37)(16,31,34,41)(17,28,18,27)(20,25,23,35)(22,38,40,44)(26,33,42,32)(30,43,39,36);(2,4)(13,16,18,33,19,34,17,32)(14,22,20,30,24,40,23,39)(15,26,28,41,29,42,27,31)(21,36,35,44,37,43,25,38)

Polycyclic presentation:

2^9|a^2=eg,b^2=ef,c^2=i,d^2=ei,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,h^c=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, 368122);