#253434

order	512
nilpotency class	3
isoclinism	isoclinic class of #253411
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	#18328	C₂
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) = 64

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

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

Polycyclic presentation:

2^9|b^2=ehi,d^2=hi,f^2=i,b^a=be,c^a=cf,d^a=de,f^a=fi,g^a=ghi,c^b=cg,f^b=fi,g^b=gh,e^c=eh,f^c=fi,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, 160630);