#6615843

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	#24128	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1603 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)(3,7)(4,8)(5,6)(17,18,22,27,23,28,21,26)(19,29,30,20,25,31,32,24);(2,6)(7,8)(9,10,12,13)(11,14,15,16)(17,19,23,25)(18,24,28,20)(21,30,22,32)(26,29,27,31);(2,6)(7,8)(10,13)(11,15)(17,20,23,24)(18,25,28,19)(21,29,22,31)(26,30,27,32);(1,3,5,4)(2,7,6,8)(9,11)(10,14)(12,15)(13,16)(17,21,23,22)(18,26,28,27)(19,30,25,32)(20,31,24,29);(1,4,5,3)(2,8,6,7)(10,13)(11,15)(17,21,23,22)(18,27,28,26)(19,30,25,32)(20,29,24,31)

Polycyclic presentation:

2^9|a^2=fg,b^2=hi,c^2=i,d^2=gi,e^2=gi,f^2=i,b^a=bf,c^a=cg,e^a=ei,c^b=ch,e^b=eh,f^b=fi,d^c=dhi,e^d=eh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72257386, #72257541, #72257600
maximal normal subgroup of	#72257386, #72257541, #72257600
maximal quotient of	#72308892

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6523039);