#468892

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

968 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)(7,11)(9,12)(14,18)(15,16)(21,22,24,28,25,29,32,27)(23,26,31,35,30,34,36,33);(5,6,8,10)(7,12,11,9)(13,14,17,18)(15,19,16,20)(21,23)(22,26)(24,31)(25,30)(27,33)(28,35)(29,34)(32,36);(5,7,8,11)(6,9,10,12)(13,15)(14,19)(16,17)(18,20)(22,27)(24,32)(26,33)(28,29)(31,36)(34,35);(2,4)(13,16,17,15)(14,20,18,19)(23,30)(26,34)(31,36)(33,35)

Polycyclic presentation:

2^9|a^2=efg,b^2=eg,c^2=g,d^2=e,f^2=i,b^a=be,c^a=cf,d^a=dh,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, 376088);