#7553747

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	#45303	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1529 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|b^2=f,c^2=f,d^2=fg,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, 7460943);