#253572

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
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

945 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ei,d^2=hi,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, 160768);