#6357952

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

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

2050 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)(3,12,14,15)(4,9,8,16)(5,10,6,13)(19,25)(20,27)(22,28)(24,30)(26,31)(29,32);(1,3,6,9,7,14,5,16)(2,8,10,12,11,4,13,15)(17,18)(19,22)(20,24)(21,23)(25,28)(26,29)(27,30)(31,32);(1,4)(2,9)(3,13)(5,12)(6,15)(7,8)(10,14)(11,16)(17,19,20,25,21,26,27,31)(18,22,24,28,23,29,30,32);(1,5,7,6)(2,10,11,13)(3,9,14,16)(4,12,8,15)(19,26)(22,29)(25,31)(28,32);(3,14)(4,8)(9,16)(12,15)(18,23)(22,29)(24,30)(28,32)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306787, #72306788, #72306789, …
maximal normal subgroup of	#72306787, #72306788, #72306789, …
maximal quotient of	#72226827, #72226828, #72226829

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265148);