#6493755

order	512
nilpotency class	3
isoclinism	isoclinic class of #6469971
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	#34609	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1631 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)(3,5)(4,6)(7,8)(11,15)(14,16)(17,18,23,31,22,24,21,25)(19,26,30,20,27,32,28,29);(1,3,4,7)(2,5,6,8)(9,10)(11,14)(12,13)(15,16)(17,19,22,27)(18,20,24,29)(21,30,23,28)(25,32,31,26);(2,6)(5,8)(10,13)(14,16)(17,20)(18,19)(21,26)(22,29)(23,32)(24,27)(25,28)(30,31);(10,13)(14,16)(19,27)(20,29)(26,32)(28,30);(9,11,12,15)(10,14,13,16)(18,24)(20,29)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6400951);