#489228

order	512
nilpotency class	4
isoclinism	isoclinic class of #488901
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	C₂×#135	D₄
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

689 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)(19,27)(20,31)(21,36)(22,24)(23,28)(25,34)(26,33)(29,47)(30,44)(32,42)(35,43)(37,41)(38,46)(39,48)(40,45);(4,13)(7,12)(9,16)(11,14)(17,19,22,30)(18,23,24,38)(20,32,35,48)(21,37,33,29)(25,41,40,47)(26,42,36,39)(27,43,44,31)(28,45,46,34);(1,3,5,10)(2,6,8,15)(4,14,13,11)(7,16,12,9)(17,18)(19,23)(20,25)(21,26)(22,24)(27,28)(29,39)(30,38)(31,34)(32,41)(33,36)(35,40)(37,42)(43,45)(44,46)(47,48);(18,24)(23,38)(25,40)(26,36)(27,44)(31,43)(39,42)(41,47)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=h,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eg,f^a=fh,g^a=gi,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

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, 396424);