#424794

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1234 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)(15,26,29,21)(16,31,17,22)(19,25,20,36)(23,32,42,34)(27,37,43,38)(28,41,33,44)(30,40,35,39);(1,2,3,4)(13,15,18,29)(14,21,24,26)(16,28,17,33)(19,35,20,30)(22,41,31,44)(23,37,42,38)(25,40,36,39)(27,34,43,32);(5,7,8,12)(6,9,10,11)(13,16,18,17)(14,22,24,31)(15,27,29,43)(19,34,20,32)(21,37,26,38)(23,25,42,36)(28,30,33,35)(39,44,40,41);(2,4)(13,17,18,16)(14,23,24,42)(15,28,29,33)(19,34,20,32)(21,38,26,37)(22,25,31,36)(27,30,43,35)(39,44,40,41)

Polycyclic presentation:

2^9|a^2=f,b^2=e,c^2=f,d^2=i,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, 331990);