#406346

order	512
nilpotency class	3
isoclinism	isoclinic class of #394469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#124
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#394
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1337 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)(9,10,13,17,14,18,27,33)(11,19,21,29,22,32,26,15)(12,16,24,30,25,31,38,40)(20,35,36,28,23,34,37,39);(1,3,4,7)(2,5,6,8)(9,11)(10,15)(12,23)(13,26)(14,22)(16,28)(17,32)(18,29)(19,33)(20,25)(21,27)(24,36)(30,35)(31,39)(34,40)(37,38);(9,12)(10,16)(11,20,22,23)(13,24)(14,25)(15,28,29,39)(17,30)(18,31)(19,34,32,35)(21,36,26,37)(27,38)(33,40);(2,6)(5,8)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=f,e^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,e^b=ei,f^b=fi,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, 313542);