#276501

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	C₂×#459	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

641 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:

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

Polycyclic presentation:

2^9|a^2=efi,d^2=ehi,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, 183697);