#6354833

order	512
nilpotency class	3
isoclinism	isoclinic class of #6354774
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

1588 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)(3,12)(4,14)(5,10)(6,13)(7,11)(8,9)(15,16)(18,22)(19,25)(20,28)(24,30)(26,27)(29,32);(1,3,6,9,7,15,5,14)(2,8,10,12,11,4,13,16)(17,18,20,22,21,24,28,30)(19,23,27,29,26,31,25,32);(1,4)(2,9)(3,13)(5,12)(6,16)(7,8)(10,15)(11,14)(17,19)(18,23)(20,27)(21,26)(22,29)(24,31)(25,28)(30,32);(1,5,7,6)(2,10,11,13)(3,14,15,9)(4,16,8,12)(18,24)(19,26)(22,30)(25,27);(1,5,7,6)(2,10,11,13)(3,9,15,14)(4,12,8,16)(18,24)(22,30)(23,31)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306850, #72306853, #72306925
maximal normal subgroup of	#72306850, #72306853, #72306925

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6262029);