#7187341

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996607
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₄⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1205 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fh,c^2=fg,d^2=f,e^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dh,e^c=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7094537);