#491528

order	512
nilpotency class	4
isoclinism	isoclinic class of #12525
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	#3147	C₂²
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

531 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)(3,9,10,16)(4,12,13,7)(6,14,15,11)(17,18)(19,28)(20,25)(21,35)(22,37)(23,27)(24,31)(26,29)(30,41)(32,47)(33,43)(34,40)(36,48)(38,45)(39,46)(42,44);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16);(3,10)(6,15)(9,16)(11,14)(17,19,29,44,22,32,36,46,23,33,34,45,20,30,21,31)(18,24,35,41,25,38,40,43,27,39,48,47,37,42,26,28);(17,20,23,22)(18,25,27,37)(19,30,33,32)(21,34,36,29)(24,38,39,42)(26,35,40,48)(28,41,43,47)(31,45,46,44)

Polycyclic presentation:

2^9|a^2=g,b^2=e,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=cg

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, 398724);