#253701

order	512
nilpotency class	3
isoclinism	isoclinic class of #253699
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
prime radical	#51040	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1006 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72233129
maximal normal subgroup of	#72233129

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 160897);