#6348114

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#120
prime radical	#60546	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2938 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)(3,9)(4,12)(5,7)(6,10)(8,14)(11,16)(13,15)(17,18)(19,25)(20,24)(21,27)(22,28)(23,26)(29,31)(30,32);(1,3,5,11)(2,6,7,15)(4,13,14,10)(8,16,12,9)(17,19,23,30)(18,24,26,31)(20,21,29,22)(25,28,32,27);(17,20)(18,25)(19,21)(22,30)(23,29)(24,28)(26,32)(27,31);(2,7)(6,15)(8,12)(9,16)(18,26)(24,31)(25,32)(27,28);(17,21,23,22)(18,27,26,28)(19,29,30,20)(24,25,31,32)

Polycyclic presentation:

2^9|b^2=h,e^2=i,f^2=h,g^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fh,g^a=gi,f^b=fh,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283748
maximal normal subgroup of	#72283748

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6255310);