#470426

order	512
nilpotency class	3
isoclinism	isoclinic class of #453981
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×C₄×C₄⋊C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

980 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)(13,14,17,16)(15,19,20,18)(21,22)(23,31)(24,30)(25,28)(26,29)(27,33)(32,36)(34,35);(5,6,8,9)(7,10,11,12)(13,15,17,20)(14,18,16,19)(21,23,25,33)(22,27,28,31)(24,34,26,32)(29,35,30,36);(2,4)(6,9)(10,12)(13,16,17,14)(15,18,20,19)(21,24)(22,28)(23,32)(25,26)(33,34)(35,36);(5,7,8,11)(6,10,9,12)(14,16)(18,19)(22,29)(24,26)(27,35)(28,30)(31,36)(32,34)

Polycyclic presentation:

2^9|a^2=ei,b^2=g,c^2=ei,d^2=eg,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^c=de,f^c=fi,h^c=hi,f^d=fi,h^d=hi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377622);