#469860

order	512
nilpotency class	3
isoclinism	isoclinic class of #453996
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

801 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 377056);