#253814

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	#3496	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

863 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)(4,8)(5,9)(6,10)(11,14)(12,15)(13,16)(17,18,23,20)(19,26,28,32)(21,31,22,25)(24,30,27,29);(1,3,6,13)(2,7,10,16)(4,12,5,11)(8,15,9,14)(17,18,23,20)(19,26)(21,31,22,25)(24,29)(27,30)(28,32);(2,8)(3,11)(4,5)(9,10)(12,13)(14,15)(17,19)(18,24)(20,27)(21,30)(22,29)(23,28)(25,32)(26,31);(17,20,23,18)(19,27,28,24)(21,25,22,31)(26,30,32,29)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,d^2=h,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, 161010);