#6358969

order	512
nilpotency class	3
isoclinism	isoclinic class of #6358842
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1595 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=eh,f^a=fh,g^a=gi,d^b=df,e^b=eh,f^b=fh,g^c=gi,f^d=fh

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283043, #72283723, #72283969
maximal normal subgroup of	#72283043, #72283723, #72283969

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6266165);