#6367381

order	512
nilpotency class	3
isoclinism	isoclinic class of #6367198
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

1628 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283124, #72283366, #72283494, …
maximal normal subgroup of	#72283124, #72283366, #72283494, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274577);