#253659

order	512
nilpotency class	3
isoclinism	isoclinic class of #253651
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	#2397	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

783 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:

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

Polycyclic presentation:

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

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, 160855);