#406774

order	512
nilpotency class	3
isoclinism	isoclinic class of #395693
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#396
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1042 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

2^9|a^2=ef,b^2=h,c^2=f,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,f^b=fi,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 313970);