#316721

order	512
nilpotency class	3
isoclinism	isoclinic class of #262915
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
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

705 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)(11,14,18,22)(12,15,20,16)(19,21,24,23)(25,26)(27,31)(28,36)(29,32)(30,33)(34,39)(35,40)(37,38);(1,3,4,8)(2,5,6,7)(9,11)(10,14)(12,19)(13,18)(15,21)(16,23)(17,22)(20,24)(25,27,29,34,30,35,28,37)(26,31,32,39,33,40,36,38);(10,15)(12,20)(14,21)(16,17)(19,24)(22,23)(25,27)(26,31)(28,34)(29,37)(30,35)(32,38)(33,40)(36,39);(2,6)(5,7)(11,18)(14,22)(19,24)(21,23)(25,28,30,29)(26,32,33,36)(27,34,35,37)(31,38,40,39)

Polycyclic presentation:

2^9|a^2=h,b^2=eg,d^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,f^a=fh,c^b=cg,d^b=dhi,f^c=fh,g^c=gi

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, 223917);