#250625

order	512
nilpotency class	3
isoclinism	isoclinic class of #250519
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	#61081	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1205 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)(11,21)(12,26)(13,14)(19,20)(22,31)(23,37)(24,25)(27,28)(29,33)(30,36)(39,40);(2,5)(3,8)(6,7)(9,10)(11,22)(12,18)(13,19)(14,16)(15,20)(17,21)(23,36)(24,32)(25,31)(26,29)(27,33)(28,34)(30,35)(37,40)(38,39);(9,11,15,25)(10,17,20,32)(12,23,28,37)(13,21,14,24)(16,22,19,31)(18,30,34,39)(26,38,27,35)(29,36,33,40);(2,5)(3,8)(6,7)(9,12,15,28)(10,18,20,34)(11,23,25,37)(13,27,14,26)(16,29,19,33)(17,30,32,39)(21,35,24,38)(22,36,31,40)

Polycyclic presentation:

2^9|a^2=hi,c^2=i,d^2=i,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gi,c^b=cg,e^b=eh,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=eh,f^d=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, 157821);