#6431933

order	512
nilpotency class	3
isoclinism	isoclinic class of #6431933
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	C₂²×#394
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3585	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4569 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72306905, #72307018, #72307052, …
maximal normal subgroup of	#72306905, #72307018, #72307052, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6339129);