#6369023

order	512
nilpotency class	3
isoclinism	isoclinic class of #6368982
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

1527 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,f^2=h,g^2=i,b^a=bf,c^a=cg,e^a=ehi,f^a=fh,g^a=gi,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	#72306613, #72306663, #72306683, …
maximal normal subgroup of	#72306613, #72306663, #72306683, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6276219);