#464890

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1048 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)(6,8)(10,11)(13,14,17,23,19,24,18,21)(15,20,25,34,28,36,27,35)(16,29,30,37,31,38,32,22)(26,40,41,43,42,44,39,33);(5,6,9,8)(7,11,12,10)(13,15,17,25,19,28,18,27)(14,20,23,34,24,36,21,35)(16,26,32,39,31,42,30,41)(22,33,38,44,37,43,29,40);(2,4)(5,7,9,12)(6,10,8,11)(14,21)(15,25)(17,18)(22,29)(23,24)(26,39)(27,28)(30,32)(34,35)(37,38)(40,44)(41,42);(5,8,9,6)(7,10,12,11)(13,16)(14,22)(15,26)(17,32)(18,30)(19,31)(20,33)(21,29)(23,38)(24,37)(25,39)(27,41)(28,42)(34,44)(35,40)(36,43)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=e,d^2=e,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,f^c=fi,g^c=gi,h^c=hi,f^d=fi,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, 372086);