#270846

order	512
nilpotency class	3
isoclinism	isoclinic class of #270811
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#62111	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

884 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 178042);