#6367715

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367534
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

1543 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283125
maximal normal subgroup of	#72283125

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274911);