#253525

order	512
nilpotency class	3
isoclinism	isoclinic class of #253507
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	C₂×C₄²	C₂⁴
prime radical	#18358	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

859 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)(3,9)(4,13)(5,8)(6,12)(7,15)(10,14)(11,16)(17,18)(19,23)(20,24)(21,25)(22,26)(27,30)(28,31)(29,32);(1,2)(3,10,11,6)(4,7)(5,8)(9,12,16,14)(13,15)(17,19,22,29)(18,23,26,32)(20,28,21,27)(24,31,25,30);(1,3)(2,6)(4,14)(5,11)(7,16)(8,10)(9,13)(12,15)(18,24)(19,27)(20,21)(25,26)(28,29)(30,31);(4,15)(7,13)(9,16)(12,14)

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160721);