#424285

order	512
nilpotency class	3
isoclinism	isoclinic class of #417647
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1513 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=ei,e^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,e^b=ei,g^c=gi

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, 331481);