#491480

order	512
nilpotency class	4
isoclinism	isoclinic class of #12506
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂×#493
prime radical	C₂³	#493
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₈² ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

508 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=eg,c^2=fh,d^2=i,e^2=g,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eg,f^a=fh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 398676);