#6459073

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437748
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	#2847	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1853 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)(3,6)(4,5)(7,8)(9,10,12,14)(11,13,15,16)(17,18)(19,26)(20,24)(21,30)(22,25)(23,29)(27,31)(28,32);(9,11)(10,13)(12,15)(14,16)(17,19,21,29,22,28,20,27)(18,23,24,26,25,31,30,32);(2,5)(6,8)(11,15)(13,16)(17,20,22,21)(18,24,25,30)(19,27,28,29)(23,26,31,32);(11,15)(13,16)(19,28)(23,31)(26,32)(27,29);(1,3,4,7)(2,6,5,8)(19,28)(23,31)(26,32)(27,29)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72256966, #72256977, #72256993
maximal normal subgroup of	#72256966, #72256977, #72256993

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6366269);