#6593089

order	512
nilpotency class	3
isoclinism	isoclinic class of #6573615
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	#32878	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2280 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2930 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257261, #72257516, #72264009, …
maximal normal subgroup of	#72257261, #72257516, #72264009, …
maximal quotient of	#72264669

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6500285);