#433737

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

881 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 340933);