#481966

order	512
nilpotency class	4
isoclinism	isoclinic class of #481819
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	#34636	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

888 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 389162);