#319580

order	512
nilpotency class	3
isoclinism	isoclinic class of #298867
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#25599	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 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)(9,10,12,14,13,15,11,16)(17,18,20,26,22,25,21,24)(19,23,27,30,28,32,29,31);(1,3,4,8)(2,5,6,7)(17,19,20,27,22,28,21,29)(18,23,26,30,25,32,24,31);(9,10)(11,14)(12,16)(13,15)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(2,6)(5,7)(9,11,13,12)(10,14,15,16)(18,25)(19,28)(24,26)(27,29)

Polycyclic presentation:

2^9|a^2=f,b^2=egi,d^2=hi,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, 226776);