#441465

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
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

898 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)(10,11)(13,14,16,18)(15,19,20,17)(21,22)(23,30)(24,35)(25,28)(26,31)(27,32)(29,33)(34,36);(5,6,8,9)(7,11,12,10)(22,28)(23,24)(25,26)(29,35)(31,32)(33,34);(13,15,16,20)(14,17,18,19)(21,23,25,33,27,34,26,24)(22,29,31,30,32,35,28,36);(2,4)(5,7,8,12)(6,10,9,11)(21,24,25,23,27,33,26,34)(22,30,31,35,32,36,28,29)

Polycyclic presentation:

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

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, 348661);