#416309

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

966 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

(5,6)(7,12)(8,10)(9,11)(13,14)(15,19)(16,20)(17,18)(21,22,24,29,25,28,32,27)(23,26,31,34,30,35,36,33);(1,2)(3,4)(5,7,9,8)(6,10,11,12)(14,18)(19,20)(21,23,25,30)(22,26,28,35)(24,31,32,36)(27,33,29,34);(2,4)(5,8,9,7)(6,10,11,12)(13,15,17,16)(14,19,18,20)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(13,16,17,15)(14,19,18,20)(22,28)(23,30)(27,29)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323505);