#476288

order	512
nilpotency class	3
isoclinism	isoclinic class of #476092
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1274 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(5,6)(7,10)(8,11)(9,12)(13,14,18,23,19,24,37,44)(15,25,28,39,29,42,32,20)(16,30,27,41,33,38,35,21)(17,34,36,43,26,22,31,40);(1,2)(3,4)(5,7)(6,9)(8,12)(10,11)(13,15)(14,20)(16,31)(17,35)(18,32)(19,29)(21,40)(22,30)(23,42)(24,39)(25,44)(26,27)(28,37)(33,36)(34,38)(41,43);(2,4)(5,7)(6,10)(8,12)(9,11)(13,16,19,33)(14,21,24,41)(15,26,29,17)(18,35,37,27)(20,22,39,34)(23,38,44,30)(25,43,42,40)(28,36,32,31);(7,12)(9,10)(13,17,19,26)(14,22,24,34)(15,27,29,35)(16,32,33,28)(18,31,37,36)(20,38,39,30)(21,25,41,42)(23,43,44,40)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=def,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=de,e^c=ei,f^c=fi,g^c=gi,h^c=hi,e^d=ei,f^d=fi,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 383484);