#325390

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

566 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eg,c^2=ef,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,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, 232586);