#7594173

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	#44974	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1485 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:

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

Polycyclic presentation:

2^9|a^2=fgh,c^2=fg,d^2=fg,e^2=fi,g^2=i,h^2=i,b^a=bf,c^a=cg,d^a=dg,e^a=ef,g^a=gi,h^a=hi,c^b=cgi,d^b=dh,g^b=gi,h^b=hi,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, 7501369);