#253914

order	512
nilpotency class	3
isoclinism	isoclinic class of #253891
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	#18405	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

875 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161110);