#268301

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	#61797	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

816 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:

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

Polycyclic presentation:

2^9|a^2=fhi,b^2=h,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, 175497);