#245844

order	512
nilpotency class	3
isoclinism	isoclinic class of #245807
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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1215 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 153040);