#6405018

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂²×C₄	C₂⁵
commutator, socle	C₂³	C₂⁴×C₄
prime radical	C₂×#120	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1444 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,16)(13,15)(17,18,23,27,22,25,30,32)(19,21,28,29,20,24,26,31);(1,3,4,8)(2,5,6,7)(9,11,13,12)(10,14,15,16)(17,19)(18,24)(20,22)(21,25)(23,28)(26,30)(27,31)(29,32);(2,6)(5,7)(9,12,13,11)(10,14,15,16)(17,20,22,19)(18,24,25,21)(23,26,30,28)(27,31,32,29);(3,8)(5,7)(18,25)(19,20)(26,28)(27,32);(2,6)(5,7)(17,21,23,29,22,24,30,31)(18,26,27,19,25,28,32,20)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ci,d^b=df,d^c=di

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6312214);