#380022

order	512
nilpotency class	3
isoclinism	isoclinic class of #196326
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#386
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1210 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

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, 287218);