#6369155

order	512
nilpotency class	3
isoclinism	isoclinic class of #6369150
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×D₄²
commutator, Frattini, cosocle	C₄²	C₂⁵
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1555 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306671, #72306683, #72306725
maximal normal subgroup of	#72306671, #72306683, #72306725

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276351);