#481316

order	512
nilpotency class	4
isoclinism	isoclinic class of #11651
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
socle	C₂²	C₂²×#102
prime radical	C₂⁴	#102
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

763 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

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

Polycyclic presentation:

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

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, 388512);