#364735

order	512
nilpotency class	3
isoclinism	isoclinic class of #196134
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1548 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=e,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271931);