#217055

order	512
nilpotency class	3
isoclinism	isoclinic class of #163298
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×C₂²⋊C₄
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

717 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,d^2=e,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=ce,d^b=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124251);