#503229

order	512
nilpotency class	4
isoclinism	isoclinic class of #503229
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#25771
commutator, Frattini, cosocle	C₂×#41	C₂⁴
LCS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂×#131 ◁ G
LECS	G ▷ C₂×#41 ▷ C₂² ▷ C₂ ▷ C₁

1123 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=e,c^2=f,d^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,g^a=ghi,c^b=cg,f^b=fh,g^b=gi,g^c=ghi,h^c=hi,g^f=gi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72237519, #72237525, #72237552, …
maximal normal subgroup of	#72237519, #72237525, #72237552, …
maximal quotient of	#72311961, #72312120, #72312686, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 410425);