#6406824

order	512
nilpotency class	2
isoclinism	isoclinic class of #42996
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×#127	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1482 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)(3,7)(4,6)(5,8)(9,10)(11,16)(12,14)(13,15)(17,18,23,21,22,25,19,26)(20,28,29,31,30,24,27,32);(1,3,4,8)(2,5,6,7)(17,19,22,23)(18,21,25,26)(20,29,30,27)(24,31,28,32);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(17,20)(18,24)(19,27)(21,32)(22,30)(23,29)(25,28)(26,31);(3,8)(5,7)(9,12,13,11)(10,14,15,16)(20,30)(24,28)(27,29)(31,32);(2,6)(5,7)(17,21,23,25,22,26,19,18)(20,31,29,24,30,32,27,28)

Polycyclic presentation:

2^9|a^2=h,b^2=f,c^2=fg,d^2=fg,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ef,c^b=ci,d^b=dfi,e^b=ei,d^c=di,e^c=ei

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6314020);