#484061

order	512
nilpotency class	4
isoclinism	isoclinic class of #484039
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#442
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1451 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

423 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=fgi,e^2=g,f^2=h,g^2=i,b^a=be,c^a=cg,d^a=di,e^a=eg,g^a=gi,e^b=egi,f^b=fi,g^b=gi,d^c=dhi,e^c=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72216790, #72216809, #72216909, …
maximal normal subgroup of	#72216790, #72216809, #72216909, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 391257);