#6367380

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
prime radical	#18682	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

1584 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)(3,11)(4,8)(5,9)(6,13)(7,10)(12,15)(14,16)(17,18,21,25)(19,26,28,31)(20,24,22,29)(23,30,32,27);(1,3)(2,5)(4,9)(6,12)(7,14)(8,16)(10,15)(11,13)(17,18)(19,23)(20,29)(21,25)(22,24)(26,27)(28,32)(30,31);(1,4,7,6)(2,8,10,13)(3,12,14,9)(5,11,15,16)(17,19,21,28)(18,23,25,32)(20,27,22,30)(24,31,29,26);(1,5,7,15)(2,9,10,12)(3,13,14,8)(4,11,6,16)(17,20,21,22)(18,24,25,29)(19,27,28,30)(23,31,32,26);(2,10)(3,14)(8,13)(9,12)(18,25)(23,32)(24,29)(26,31)

Polycyclic presentation:

2^9|a^2=i,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, #72283531, #72283800, …
maximal normal subgroup of	#72283124, #72283531, #72283800, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6274576);