#216957

order	512
nilpotency class	3
isoclinism	isoclinic class of #162948
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

729 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)(11,16)(12,21)(13,18)(14,23)(15,19)(17,22)(20,24)(25,26,29,32,30,34,28,33)(27,31,36,38,37,40,35,39);(1,2)(3,6)(4,5)(7,8)(9,11,15,12)(10,16,19,21)(13,22,14,20)(17,23,24,18)(25,27,30,37)(26,31,34,40)(28,35,29,36)(32,38,33,39);(2,5)(6,8)(9,12,15,11)(10,17,19,24)(13,22,14,20)(16,23,21,18)(26,32)(28,29)(31,38)(33,34)(35,36)(39,40);(1,3,4,7)(2,6,5,8)(9,13,15,14)(10,18,19,23)(11,20,12,22)(16,24,21,17)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 124153);