#449420

order	512
nilpotency class	3
isoclinism	isoclinic class of #449419
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15943
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

726 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)(4,9)(5,20,23,14)(6,11)(8,12)(10,22,21,24)(13,18)(15,19)(16,27,31,29)(17,26)(25,32,28,30);(1,3,7,2,8,11,6,12)(4,15,18,13,19,9,17,26)(5,21,22,20,23,10,24,14)(16,29,30,25,31,27,32,28);(1,4,7,18,8,19,6,17)(2,9,3,13,12,15,11,26)(5,16,24,32,23,31,22,30)(10,25,14,27,21,28,20,29);(1,5)(2,10,12,21)(3,14,11,20)(4,16)(6,22)(7,24)(8,23)(9,25,15,28)(13,27,26,29)(17,30)(18,32)(19,31)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72251684, #72251693, #72251700, …
maximal normal subgroup of	#72251684, #72251693, #72251700, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 356616);