#432270

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

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

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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=ef,d^2=f,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=dei,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, 339466);