#474823

order	512
nilpotency class	3
isoclinism	isoclinic class of #454375
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

866 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)(7,11)(9,12)(13,14)(15,17)(16,18)(19,20)(21,22)(23,32)(24,28)(25,30)(26,31)(27,33)(29,34)(35,36);(5,6,8,10)(7,12,11,9)(13,15,16,19)(14,17,18,20)(21,23,25,34,26,35,24,33)(22,27,28,36,31,29,30,32);(2,4)(5,7,8,11)(6,9,10,12)(22,28)(23,33)(24,25)(30,31)(32,36)(34,35);(14,18)(17,20)(21,23,25,34,26,35,24,33)(22,29,28,32,31,27,30,36)

Polycyclic presentation:

2^9|a^2=fg,b^2=egh,c^2=eg,d^2=ei,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi

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, 382019);