#6458745

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437753
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1946 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6365941);