#424313

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=ef,d^2=ei,e^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,e^b=ei,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, 331509);