#401763

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	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1370 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=i,d^2=ei,f^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=di,f^c=fi

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, 308959);