#316795

order	512
nilpotency class	3
isoclinism	isoclinic class of #262675
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

616 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 223991);