#254373

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

1080 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:

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

Polycyclic presentation:

2^9|b^2=e,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.

Relationships

maximal subgroup of	#72237004
maximal normal subgroup of	#72237004

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 161569);