#254428

order	512
nilpotency class	3
isoclinism	isoclinic class of #254419
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

1068 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)(11,19)(16,22)(20,23)(21,28)(25,27)(29,31);(2,9)(3,14,16,24)(6,12)(10,22,26,13)(11,17)(15,27,30,31)(18,25,32,29)(23,28);(1,3)(2,10)(4,18)(5,22)(6,14)(7,16)(8,13)(9,24)(11,27)(12,26)(15,21)(17,29)(19,30)(20,32)(23,25)(28,31);(1,4)(2,11)(3,15)(5,19)(6,23)(7,20)(8,21)(9,17)(10,25)(12,28)(13,18)(14,27)(16,30)(22,32)(24,31)(26,29)

Polycyclic presentation:

2^9|b^2=ehi,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,d^c=di,e^c=eh,f^c=fi,g^c=gi,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, 161624);