#6355375

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355367
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#121
commutator, Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1664 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)(4,8)(5,14)(6,9)(7,13)(11,16)(12,15)(18,23)(19,27)(20,21)(25,26)(28,29)(30,31);(1,3,5,13,6,11,4,12)(2,7,8,10,9,15,14,16)(17,18,21,23,22,26,20,25)(19,24,28,30,29,32,27,31);(17,19,22,29)(18,24,26,32)(20,27,21,28)(23,30,25,31);(3,11)(7,15)(10,16)(12,13);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)(17,20,22,21)(18,25,26,23)(19,27,29,28)(24,31,32,30)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283531, #72306988, #72306999, …
maximal normal subgroup of	#72283531, #72306988, #72306999, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262571);