#250187

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	#61097	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1199 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)(10,16)(12,26)(13,14)(17,21)(19,20)(22,36)(23,24)(27,28)(29,33)(30,39)(31,32)(37,38);(2,5)(3,8)(6,7)(9,10,15,20)(11,21,25,31)(12,18,27,34)(13,19,14,16)(17,23,32,24)(22,35,37,40)(26,29,28,33)(30,38,39,36);(9,11,15,25)(10,17,20,32)(12,22,27,37)(13,24,14,23)(16,21,19,31)(18,30,34,39)(26,38,28,36)(29,35,33,40);(2,5)(3,8)(6,7)(9,12)(10,18)(11,22)(13,28)(14,26)(15,27)(16,29)(17,30)(19,33)(20,34)(21,35)(23,38)(24,36)(25,37)(31,40)(32,39)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^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, 157383);