#323975

order	512
nilpotency class	3
isoclinism	isoclinic class of #278923
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

688 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)(3,7)(4,6)(5,8)(17,18,20,23)(19,25,27,32)(21,29,26,24)(22,30,31,28);(1,3,4,8)(2,5,6,7)(9,10,11,14,12,15,16,13)(17,18)(19,22)(20,23)(21,24)(25,28)(26,29)(27,31)(30,32);(10,13)(11,16)(14,15)(17,19,26,30,20,27,21,28)(18,22,29,32,23,31,24,25);(2,6)(5,7)(18,23)(22,31)(24,29)(25,32)

Polycyclic presentation:

2^9|a^2=h,b^2=egh,c^2=fh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,c^b=cg,d^b=dh,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 231171);