#7531569

order	512
nilpotency class	3
isoclinism	isoclinic class of #7514040
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35728	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1890 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=fi,d^2=fi,e^2=f,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,e^b=ei,h^b=hi,e^c=ei,h^d=hi

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7438765);