#6385620

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	#2714	C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1523 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=fg,d^2=f,e^2=h,h^2=i,b^a=bf,c^a=cg,e^a=ei,d^b=dfi,d^c=di,e^c=ei,e^d=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, 6292816);