#347441

order	512
nilpotency class	3
isoclinism	isoclinic class of #347434
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2238 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)(10,16)(11,21)(12,24)(13,14)(18,33)(19,20)(23,27)(25,26)(28,31)(30,38)(35,36)(39,40);(3,7)(5,8)(9,10)(11,17)(12,18)(13,19)(14,16)(15,20)(21,31)(22,30)(23,32)(24,29)(25,34)(26,33)(27,28)(35,39)(36,38)(37,40);(1,2,4,6)(3,8,7,5)(9,11)(10,17)(12,22)(13,21)(14,27)(15,23)(16,28)(18,30)(19,31)(20,32)(24,36)(25,35)(26,37)(29,38)(33,40)(34,39);(9,12,15,26)(10,18,20,33)(11,22,23,37)(13,25,14,24)(16,29,19,34)(17,30,32,40)(21,35,27,36)(28,38,31,39)

Polycyclic presentation:

2^9|c^2=i,d^2=h,e^2=h,f^2=hi,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fhi,g^a=gh,e^b=eh,f^b=fi,e^c=ei,f^c=fhi,g^d=gh

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254637);