#253480

order	512
nilpotency class	3
isoclinism	isoclinic class of #253459
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	#18316	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

918 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)(3,11)(4,10)(5,9)(6,8)(7,13)(12,16)(14,15)(19,25)(20,28)(22,31)(23,29)(26,32)(27,30);(1,3)(2,7)(4,12)(5,13)(6,14)(8,16)(9,11)(10,15)(17,18)(19,26,27,22)(20,29)(21,24)(23,28)(25,31,30,32);(2,8)(4,6)(7,15)(9,10)(11,16)(12,14)(17,19)(18,22)(20,30)(21,27)(23,32)(24,26)(25,28)(29,31);(1,4,5,6)(2,8,9,10)(3,12,13,14)(7,16,11,15)(19,27)(20,28)(22,26)(23,29)

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,d^c=di,e^c=eh,f^c=fi,f^d=fh

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160676);