#6357814

order	512
nilpotency class	3
isoclinism	isoclinic class of #6357798
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₈

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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=g,d^2=h,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, #72224737
maximal normal subgroup of	#72219655, #72224737
maximal quotient of	#72224725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6265010);