#279399

order	512
nilpotency class	3
isoclinism	isoclinic class of #267131
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

648 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 186595);