#6355412

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355407
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
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1677 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)(3,6)(4,7)(5,8)(9,12)(10,14)(11,13)(15,16)(18,22)(19,25)(20,28)(24,30)(26,27)(29,31);(1,3)(2,6)(4,11)(5,10)(7,13)(8,14)(9,15)(12,16)(17,18,20,22,21,24,28,30)(19,23,27,29,26,32,25,31);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,19)(18,23)(20,27)(21,26)(22,29)(24,32)(25,28)(30,31);(3,10)(6,14)(11,16)(13,15)(18,24)(19,26)(22,30)(25,27);(2,8)(6,14)(7,9)(13,15)(17,20,21,28)(18,22,24,30)(19,27,26,25)(23,29,32,31)

Polycyclic presentation:

2^9|b^2=f,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=di,d^c=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306812, #72306814, #72306819, …
maximal normal subgroup of	#72306812, #72306814, #72306819, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262608);