#254382

order	512
nilpotency class	3
isoclinism	isoclinic class of #254371
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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1071 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161578);