#359812

order	512
nilpotency class	3
isoclinism	isoclinic class of #196338
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

2276 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)(11,15)(12,22,24,16)(13,17)(14,18)(19,26,33,29)(20,27)(21,28)(23,35,36,30)(25,31)(32,37,40,39)(34,38);(2,5)(6,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,29)(17,27)(18,28)(22,26)(23,32)(24,33)(25,34)(30,39)(31,38)(35,37)(36,40);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,23)(14,24)(15,26)(17,30)(18,22)(20,32)(21,33)(25,36)(27,37)(28,29)(31,35)(34,40)(38,39);(2,5)(6,8)(9,13,14,25)(10,17,18,31)(11,20,21,34)(12,23,24,36)(15,27,28,38)(16,30,22,35)(19,32,33,40)(26,37,29,39)

Polycyclic presentation:

2^9|a^2=h,c^2=e,d^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=gi,c^b=cg,f^b=fi,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, 267008);