#451628

order	512
nilpotency class	3
isoclinism	isoclinic class of #451628
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#16005
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

799 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)(3,13,16,8)(4,17)(5,12)(6,10)(7,11)(9,19)(14,23,30,29)(15,25,22,24)(18,32,31,26)(20,28)(21,27);(1,3)(2,8)(4,18,19,31)(5,15)(6,22)(7,16)(9,26,17,32)(10,25)(11,13)(12,24)(14,20,30,21)(23,28,29,27);(1,4)(2,9,11,17)(3,14)(5,21)(6,20)(7,19)(8,23,13,29)(10,27,12,28)(15,31)(16,30)(18,22)(24,26,25,32);(2,10)(5,6)(8,24)(9,27)(11,12)(13,25)(15,22)(17,28)(18,31)(20,21)(23,26)(29,32)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251785, #72251789, #72251797, …
maximal normal subgroup of	#72251785, #72251789, #72251797, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358824);