#278628

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	#3038	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

524 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

2^9|a^2=fh,b^2=e,c^2=i,d^2=ei,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, 185824);