#6357141

order	512
nilpotency class	3
isoclinism	isoclinic class of #19621
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₄²	C₂⁵
socle	C₂²	C₂²×#127
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2131 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)(3,10)(4,9)(5,14)(6,8)(7,13)(11,16)(12,15)(17,18)(19,26)(20,24)(21,30)(22,25)(23,28)(27,31)(29,32);(1,3,5,13,6,11,4,12)(2,7,9,10,8,15,14,16);(2,8)(7,15)(9,14)(10,16)(17,19,21,28,22,29,20,27)(18,23,24,26,25,31,30,32);(3,11)(7,15)(10,16)(12,13);(1,4,6,5)(2,9,8,14)(3,12,11,13)(7,10,15,16)(17,20,22,21)(18,24,25,30)(19,27,29,28)(23,26,31,32)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72285524, #72285558, #72285618, …
maximal normal subgroup of	#72285524, #72285558, #72285618, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6264337);