#7043284

order	512
nilpotency class	3
isoclinism	isoclinic class of #6996335
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	C₂×#466	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×C₂²⋊Q₈ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1450 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=g,d^2=f,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313419, #72313674, #72313734
maximal normal subgroup of	#72313419, #72313674, #72313734

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6950480);