#469755

order	512
nilpotency class	3
isoclinism	isoclinic class of #453966
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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 ◁ C₂×#376 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1156 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)(10,12)(14,17)(18,19)(21,22,25,28)(23,30,32,27)(24,33,35,36)(26,31,34,29);(5,6,8,9)(7,10,11,12)(13,14,16,17)(15,19,20,18)(21,23,25,32)(22,27,28,30)(24,29,35,31)(26,33,34,36);(7,11)(10,12)(13,15)(14,18)(16,20)(17,19)(21,24,34,30,25,35,26,27)(22,23,31,36,28,32,29,33);(2,4)(5,7,8,11)(6,10,9,12)(15,20)(18,19)(23,32)(24,35)(27,30)(33,36)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,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

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, 376951);