#250148

order	512
nilpotency class	3
isoclinism	isoclinic class of #250071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3499	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1057 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 157344);