#474423

order	512
nilpotency class	3
isoclinism	isoclinic class of #454382
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×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 ◁ C₄×#95 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

878 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)(5,6,8,10)(7,9,11,12)(14,17)(18,20)(21,22,26,30)(23,31,34,35)(24,29,25,28)(27,33,36,32);(7,11)(9,12)(13,14)(15,18)(16,17)(19,20)(21,23,24,32,26,34,25,33)(22,27,28,31,30,36,29,35);(2,4)(5,7,8,11)(6,9,10,12)(14,17)(15,19)(23,32)(24,25)(27,31)(28,29)(33,34)(35,36);(13,15,16,19)(14,18,17,20)(21,23,25,33,26,34,24,32)(22,27,29,35,30,36,28,31)

Polycyclic presentation:

2^9|a^2=egi,b^2=h,c^2=eg,d^2=ei,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=di,d^c=de,e^c=ei,g^c=gi,h^c=hi

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, 381619);