#434595

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1092 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=e,c^2=f,d^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^b=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, 341791);