#7616338

order	512
nilpotency class	3
isoclinism	isoclinic class of #7518944
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	#2397	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1399 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)(4,5)(6,8)(11,12)(14,15)(19,26)(24,27)(28,32)(29,31);(2,5)(6,7)(9,10,13,16)(11,15,12,14)(17,18,22,25)(19,27,26,24)(20,23,30,21)(28,31,32,29);(9,11,13,12)(10,14,16,15)(17,19,22,26)(18,24,25,27)(20,28,30,32)(21,31,23,29);(9,12,13,11)(10,15,16,14)(17,20,23,25,22,30,21,18)(19,28,29,27,26,32,31,24);(1,3,4,8)(2,6,5,7)(17,21,22,23)(18,20,25,30)(19,29,26,31)(24,32,27,28)

Polycyclic presentation:

2^9|b^2=g,c^2=g,d^2=h,e^2=fi,h^2=i,b^a=bf,c^a=cg,d^a=dgi,e^a=ef,c^b=cg,d^b=dh,h^b=hi,e^c=ei,e^d=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, 7523534);