#6722406

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667580
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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1661 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:

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

Polycyclic presentation:

2^9|a^2=i,c^2=g,d^2=f,e^2=g,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efhi,h^b=hi,d^c=di,h^c=hi,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257411, #72257553, #72257681
maximal normal subgroup of	#72257411, #72257553, #72257681
maximal quotient of	#72308550

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6629602);