#6355578

order	512
nilpotency class	3
isoclinism	isoclinic class of #6355552
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	#60917	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1539 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=hi,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,e^b=eh,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72307016, #72307019, #72307064
maximal normal subgroup of	#72307016, #72307019, #72307064

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262774);