#319480

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

838 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)(17,18,20,24,21,23,29,31)(19,25,26,30,27,32,28,22);(1,3,4,8)(2,5,6,7)(9,10,11,15,12,14,16,13);(10,13)(11,16)(14,15)(17,19,21,27)(18,22,23,30)(20,28,29,26)(24,32,31,25);(2,6)(5,7)(10,14)(13,15)(18,23)(22,30)(24,31)(25,32)

Polycyclic presentation:

2^9|a^2=f,b^2=eg,c^2=h,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=deh,c^b=cg,d^b=di,f^c=fh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 226676);