#415586

order	512
nilpotency class	3
isoclinism	isoclinic class of #393889
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1192 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)(15,19)(16,17)(18,20)(21,22,26,27,25,30,24,29)(23,28,33,34,31,35,32,36);(5,6,8,10)(7,11,12,9)(14,17)(18,19)(22,27)(24,26)(28,34)(29,30)(32,33)(35,36);(2,4)(5,7,8,12)(6,9,10,11)(21,23,25,31)(22,28,30,35)(24,32,26,33)(27,34,29,36);(13,15,16,20)(14,18,17,19)(23,31)(28,35)(32,33)(34,36)

Polycyclic presentation:

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

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, 322782);