#352675

order	512
nilpotency class	3
isoclinism	isoclinic class of #352019
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

690 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)(10,14)(11,17)(12,20)(15,21)(16,18)(19,23)(25,26,29,32)(27,33,36,40)(28,37,34,31)(30,38,39,35);(2,5)(6,8)(9,10,12,14,13,15,20,21)(11,18,17,24,19,16,23,22);(10,15)(14,21)(16,18)(22,24)(25,27,34,38,29,36,28,35)(26,30,37,40,32,39,31,33);(1,3,4,7)(2,6,5,8)(9,11,12,17,13,19,20,23)(10,16,14,22,15,18,21,24)

Polycyclic presentation:

2^9|a^2=i,b^2=g,c^2=fi,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=ch,d^b=dh

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259871);