#414061

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

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

996 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:

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

Polycyclic presentation:

2^9|a^2=e,b^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, 321257);