#250119

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	#61094	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1230 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,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, 157315);