#483098

order	512
nilpotency class	4
isoclinism	isoclinic class of #481466
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#102
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#60718	C₂
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₄×#41 ◁ C₄×#131 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

739 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=i,e^2=hi,f^2=ghi,h^2=i,b^a=be,e^a=eg,e^b=eh,f^b=fgh,h^b=hi,d^c=di

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 390294);