#6401728

order	512
nilpotency class	2
isoclinism	isoclinic class of #35680
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	C₂²×D₄	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1464 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)(9,10)(11,14)(12,13)(15,16)(17,18,23,21,22,25,20,26)(19,27,29,32,28,24,31,30);(1,3,4,7)(2,5,6,8)(10,13)(11,15)(17,19)(18,24)(20,31)(21,30)(22,28)(23,29)(25,27)(26,32);(2,6)(5,8)(10,13)(14,16)(18,25)(19,28)(21,26)(29,31);(9,11,12,15)(10,14,13,16)(17,20,22,23)(18,26,25,21)(19,29,28,31)(24,30,27,32);(10,13)(14,16)(17,21,23,25,22,26,20,18)(19,30,29,27,28,32,31,24)

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=ef,c^b=ci,d^b=df,e^c=ei

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6308924);