#409982

order	512
nilpotency class	3
isoclinism	isoclinic class of #394040
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1571 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)(11,21)(12,26)(13,18)(14,19)(15,20)(16,24)(17,29)(22,37)(23,33)(25,32)(27,40)(28,36)(30,35)(31,39)(34,38);(1,3,4,7)(2,5,6,8)(9,11)(10,16)(12,27)(13,23)(14,24)(15,25)(17,34)(18,21)(19,32)(20,33)(22,29)(26,37)(28,39)(30,38)(31,35)(36,40);(9,12,14,29,15,30,13,28)(10,17,19,35,20,36,18,26)(11,22,23,27,25,39,24,38)(16,31,21,34,33,37,32,40);(2,6)(5,8)(9,13,15,14)(10,18,20,19)(11,23,25,24)(12,28,30,29)(16,21,33,32)(17,26,36,35)(22,27,39,38)(31,34,37,40)

Polycyclic presentation:

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

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, 317178);