#253818

order	512
nilpotency class	3
isoclinism	isoclinic class of #253795
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	C₂³×C₄	C₂⁴
prime radical	#3505	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

801 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161014);