#6444396

order	512
nilpotency class	3
isoclinism	isoclinic class of #6444396
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂×#2428
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2638 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=f,c^2=fg,d^2=f,f^2=i,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,f^b=fi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213913, #72214632, #72219485, …
maximal normal subgroup of	#72213913, #72214632, #72219485, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6351592);