#253353

order	512
nilpotency class	3
isoclinism	isoclinic class of #253339
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3449	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

819 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)(3,7)(4,9)(5,8)(6,10)(11,14)(12,16)(13,15)(17,18,22,26)(19,23,21,24)(20,28,31,27)(25,30,32,29);(1,3)(2,7)(4,11)(5,13)(6,12)(8,15)(9,14)(10,16)(17,19,22,21)(18,24,26,23)(20,25)(27,29)(28,30)(31,32);(2,8)(4,5)(7,14)(9,10)(11,13)(15,16)(17,20)(18,25)(19,27)(21,28)(22,31)(23,30)(24,29)(26,32);(1,4,6,5)(2,9,10,8)(3,11,12,13)(7,14,16,15)(17,21,22,19)(18,23,26,24)(20,27,31,28)(25,29,32,30)

Polycyclic presentation:

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

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, 160549);