#352854

order	512
nilpotency class	3
isoclinism	isoclinic class of #352174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

606 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,c^2=f,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,d^b=dhi,d^c=dh

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, 260050);