#268284

order	512
nilpotency class	3
isoclinism	isoclinic class of #268283
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	#61673	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

910 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:

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

Polycyclic presentation:

2^9|a^2=fh,e^2=i,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,e^b=ei,f^b=fhi,g^b=gi,e^c=ehi,f^c=fh,g^c=gi,e^d=ei,f^d=fhi

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, 175480);