#243877

order	512
nilpotency class	3
isoclinism	isoclinic class of #232809
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#399
prime radical	C₂×#121	Q₈
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

693 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151073);