#469824

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

1018 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)(10,11)(21,22,26,29)(23,27,31,34)(24,33,28,25)(30,36,35,32);(5,6,8,9)(7,11,12,10)(13,14)(15,18)(16,17)(19,20)(21,23)(22,27)(24,30)(25,32)(26,31)(28,35)(29,34)(33,36);(5,7,8,12)(6,10,9,11)(14,17)(18,20)(21,24,25,22,26,28,33,29)(23,30,32,27,31,35,36,34);(2,4)(7,12)(10,11)(13,15,16,19)(14,18,17,20)(23,31)(27,34)(30,35)(32,36)

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=ef,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, 377020);