#438299

order	512
nilpotency class	3
isoclinism	isoclinic class of #417778
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

1168 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)(6,7)(9,10)(11,22)(12,17)(13,31)(14,19)(15,20)(16,26)(18,29)(21,23)(24,33)(25,36)(27,35)(28,40)(30,37)(32,39)(34,38);(5,6)(7,8)(9,11,23,38,14,26,15,27)(10,16,21,35,19,22,20,34)(12,28,30,24,13,25,29,39)(17,36,37,32,31,40,18,33);(1,2,3,4)(5,6,8,7)(9,12,14,13)(10,17,19,31)(11,24,26,39)(15,30,23,29)(16,32,22,33)(18,20,37,21)(25,38,28,27)(34,40,35,36);(5,7,8,6)(9,13,14,12)(10,18,19,37)(11,25,26,28)(15,30,23,29)(16,33,22,32)(17,20,31,21)(24,38,39,27)(34,40,35,36)

Polycyclic presentation:

2^9|b^2=eg,c^2=fi,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,g^c=gi,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, 345495);