#319366

order	512
nilpotency class	3
isoclinism	isoclinic class of #298795
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

821 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:

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

Polycyclic presentation:

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

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, 226562);