#471524

order	512
nilpotency class	3
isoclinism	isoclinic class of #455107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1097 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,18)(19,28)(20,33)(21,25)(22,27)(23,31)(24,36)(26,29)(30,41)(32,39)(34,44)(35,43)(37,42)(38,47)(40,45)(46,48);(2,5)(6,8)(9,11,12,15)(10,13,14,16)(17,19,29,45,21,31,22,32)(18,23,26,39,25,28,27,40)(20,34,37,30,35,47,36,46)(24,41,33,38,42,48,43,44);(3,7)(6,8)(17,20)(18,24)(19,30)(21,35)(22,37)(23,38)(25,42)(26,43)(27,33)(28,44)(29,36)(31,46)(32,47)(34,45)(39,41)(40,48);(1,3,4,7)(2,6,5,8)(10,14)(13,16)(18,25)(20,35)(23,28)(26,27)(30,46)(34,47)(36,37)(39,40)

Polycyclic presentation:

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

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, 378720);