#278612

order	512
nilpotency class	3
isoclinism	isoclinic class of #278603
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

503 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=e,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,f^c=fi,g^c=ghi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 185808);