#414511

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

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

806 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)(3,8)(4,6)(5,7)(17,18,24,27)(19,28,20,31)(21,22,26,23)(25,30,32,29);(1,3,5,4)(2,6,7,8)(9,10)(11,14)(12,13)(15,16)(17,19)(18,25)(20,24)(21,31)(22,30)(23,29)(26,28)(27,32);(1,4,5,3)(2,6,7,8)(10,13)(14,16)(17,20)(18,25)(19,24)(21,31)(22,29)(23,30)(26,28)(27,32);(9,11,12,15)(10,14,13,16)(17,21,23,18,24,26,22,27)(19,28,29,32,20,31,30,25)

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=ef,d^2=gh,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,d^b=di,e^b=ei,f^b=fi,e^c=ei,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, 321707);