#470253

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

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

1288 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)(7,10)(8,9)(11,12)(14,17)(18,19)(21,22,25,29)(23,30,32,36)(24,28,26,34)(27,31,35,33);(6,9)(10,12)(13,14,16,17)(15,19,20,18)(21,22)(23,27)(24,34)(25,29)(26,28)(30,33)(31,36)(32,35);(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,23,25,32)(22,27,29,35)(24,33,26,31)(28,36,34,30);(2,4)(5,7,8,11)(6,10,9,12)(15,20)(18,19)(21,24,25,26)(22,28,29,34)(23,31,32,33)(27,30,35,36)

Polycyclic presentation:

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

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, 377449);