#491641

order	512
nilpotency class	4
isoclinism	isoclinic class of #491572
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	#33773	C₂
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₈×M₁₆ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

535 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)(3,9)(4,12)(5,7)(6,11)(8,13)(10,15)(14,16)(17,18,21,24)(19,28,31,43)(20,26,23,37)(22,36,35,48)(25,40,38,32)(27,34,39,29)(30,44,33,45)(41,47,42,46);(1,3,4,11,5,10,13,16)(2,6,8,9,7,14,12,15)(18,24)(25,38)(26,37)(27,39)(28,43)(36,48)(41,42)(44,45);(2,7)(6,14)(8,12)(9,15)(17,19,29,46,23,30,35,40,21,31,34,47,20,33,22,32)(18,25,36,45,26,42,39,43,24,38,48,44,37,41,27,28);(3,10)(6,14)(9,15)(11,16)(17,20,21,23)(18,26,24,37)(19,30,31,33)(22,34,35,29)(25,41,38,42)(27,36,39,48)(28,44,43,45)(32,46,40,47)

Polycyclic presentation:

2^9|a^2=i,b^2=ei,c^2=fgh,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,d^c=di

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, 398837);