#470120

order	512
nilpotency class	3
isoclinism	isoclinic class of #453960
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×C₄³ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 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:

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

Polycyclic presentation:

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

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, 377316);