#285668

order	512
nilpotency class	3
isoclinism	isoclinic class of #281531
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

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

656 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)(17,18,22,24,21,25,20,26)(19,27,31,28,30,23,29,32);(2,5)(6,8)(10,13)(11,15)(14,16)(17,19,20,29,21,30,22,31)(18,23,26,28,25,27,24,32);(1,3,4,7)(2,6,5,8)(9,10,12,14)(11,16,15,13)(17,18,21,25)(19,28,30,32)(20,24,22,26)(23,31,27,29);(2,5)(6,8)(17,20,21,22)(18,24,25,26)(19,29,30,31)(23,32,27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 192864);