#6456014

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437510
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#26939	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1652 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)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18,23,20)(19,24,29,30)(21,25,22,26)(27,31,32,28);(2,5)(6,8)(17,19)(18,24)(20,30)(21,28)(22,31)(23,29)(25,27)(26,32);(10,13)(14,16)(17,20,23,18)(19,27,29,32)(21,25,22,26)(24,31,30,28);(1,3,4,7)(2,6,5,8)(17,21)(18,25)(20,26)(22,23)(27,32)(28,31);(9,11,12,15)(10,14,13,16)(17,21,23,22)(18,25,20,26)(19,28,29,31)(24,27,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363210);