#450130

order	512
nilpotency class	3
isoclinism	isoclinic class of #450130
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15958
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1048 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)(3,7,13,22)(4,16,18,9)(5,11)(6,8)(10,12)(14,21)(15,23,26,20)(17,29)(19,25,27,24)(28,32)(30,31);(1,3,12,26,5,13,6,15)(2,7,10,20,11,22,8,23)(4,17,27,28,18,14,19,30)(9,21,25,31,16,29,24,32);(2,8)(3,13)(6,12)(7,20)(9,24)(10,11)(14,17)(16,25)(19,27)(21,31)(22,23)(29,32);(1,4,5,18)(2,9)(3,14,13,17)(6,19,12,27)(7,21)(8,24)(10,25)(11,16)(15,28,26,30)(20,31)(22,29)(23,32)

Polycyclic presentation:

2^9|a^2=eh,b^2=ef,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=de,g^b=gi,h^b=hi,f^c=fi,h^d=hi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72319564
maximal normal subgroup of	#72319564

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 357326);