#491588

order	512
nilpotency class	4
isoclinism	isoclinic class of #491556
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#493
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

502 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)(21,34)(22,37)(23,27)(24,31)(26,35)(29,41)(30,42)(32,48)(33,44)(36,38)(39,45)(40,47)(43,46);(1,3,4,11,5,10,13,14)(2,6,7,9,8,15,12,16);(17,19,29,45,22,32,21,31,23,33,35,46,20,30,36,47)(18,24,38,48,25,39,26,28,27,40,34,42,37,43,41,44);(3,10)(6,15)(9,16)(11,14)(17,20,23,22)(18,25,27,37)(19,30,33,32)(21,29,36,35)(24,39,40,43)(26,34,41,38)(28,42,44,48)(31,45,47,46)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=fhi,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,d^b=dg

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, 398784);