#253794

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
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1042 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=ehi,d^2=hi,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, 160990);