#6366583

order	512
nilpotency class	3
isoclinism	isoclinic class of #6366542
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

1633 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72338415, #72338434, #72338443
maximal normal subgroup of	#72338415, #72338434, #72338443

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6273779);