#414941

order	512
nilpotency class	3
isoclinism	isoclinic class of #394465
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

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×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1295 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=efh,d^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,f^b=fi,g^b=gi,e^c=ei,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, 322137);