#286148

order	512
nilpotency class	3
isoclinism	isoclinic class of #282043
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

635 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ei,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=deh,c^b=cg,g^b=gi,f^c=fh,g^c=ghi

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 193344);