#472502

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

970 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:

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

Polycyclic presentation:

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

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, 379698);