#278646

order	512
nilpotency class	3
isoclinism	isoclinic class of #278635
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2416
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#41	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

505 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)(3,8)(4,14)(5,12)(6,11)(7,10)(9,15)(13,16)(17,18,23,25,22,20,21,26)(19,27,31,30,29,32,28,24);(1,3,6,12,7,4,15,13)(2,5,9,8,10,16,11,14)(17,19,21,28,22,29,23,31)(18,24,25,27,20,30,26,32);(1,4)(2,8)(3,7)(5,11)(6,13)(9,16)(10,14)(12,15)(17,20,22,18)(19,24,29,30)(21,26,23,25)(27,31,32,28);(1,5,15,14,7,16,6,8)(2,3,11,13,10,4,9,12)(17,19,23,31,22,29,21,28)(18,24,26,32,20,30,25,27)

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=eh,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^b=dh,g^b=ghi,d^c=dh,f^c=fi,g^c=ghi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185842);