#6357811

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357798
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

2070 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)(17,18)(19,25)(20,22)(21,29)(23,27)(24,28)(26,31)(30,32);(1,3,4,12,6,11,5,13)(2,7,14,16,9,15,8,10)(18,22)(23,30)(24,29)(25,31);(1,2)(3,7)(4,14)(5,8)(6,9)(10,13)(11,15)(12,16)(17,19,21,27,20,26,28,32)(18,23,24,25,22,30,29,31);(3,11)(7,15)(10,16)(12,13)(19,26)(23,30)(25,31)(27,32);(1,4,6,5)(2,8,9,14)(3,12,11,13)(7,10,15,16)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72219655, #72306878, #72306928, …
maximal normal subgroup of	#72219655, #72306878, #72306928, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265007);