#445018

order	512
nilpotency class	3
isoclinism	isoclinic class of #445018
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15797
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

802 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 352214);