#414223

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

838 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=e,c^2=e,d^2=gh,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,h^a=hi,c^b=cg,d^b=di,f^c=fi

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, 321419);