#443904

order	512
nilpotency class	3
isoclinism	isoclinic class of #443903
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂	#15768
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

758 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:

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

Polycyclic presentation:

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

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, 351100);