#6590751

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573523
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	#2433	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1628 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)(11,14)(12,16)(13,15)(17,18)(19,24)(20,21)(22,25)(23,28)(26,27)(29,32)(30,31);(1,3,5,4)(2,6,8,7)(9,11,13,12)(10,14,15,16)(17,19,21,28)(18,23,20,24)(22,27,31,29)(25,32,30,26);(1,3,5,4)(2,7,8,6)(10,15)(14,16)(17,20,21,18)(19,26,28,32)(22,25,31,30)(23,27,24,29);(9,12,13,11)(10,16,15,14)(17,19,21,28)(18,24,20,23)(22,27,31,29)(25,26,30,32);(1,4,5,3)(2,6,8,7)(18,20)(19,27)(22,31)(23,26)(24,32)(28,29)

Polycyclic presentation:

2^9|a^2=fi,b^2=fgi,c^2=f,d^2=g,e^2=fi,h^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,h^b=hi,d^c=dh,e^c=ei,h^c=hi,e^d=eh,h^d=hi,h^e=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72284905, #72285539
maximal normal subgroup of	#72284905, #72285539

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6497947);