#245876

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

1186 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:

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

Polycyclic presentation:

2^9|b^2=i,c^2=h,d^2=hi,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, 153072);