#482560

order	512
nilpotency class	4
isoclinism	isoclinic class of #482362
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

897 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)(19,29)(20,25)(21,37,35,26)(22,38,39,27)(23,28)(24,32,42,46)(30,41)(31,43,47,40)(33,45)(34,48)(36,44);(1,3)(2,7)(4,12)(5,14)(6,13)(8,15)(9,11)(10,16)(17,19,23,33)(18,24,28,42)(20,34,36,30)(21,31,35,47)(22,32,39,46)(25,43,44,40)(26,41,37,48)(27,29,38,45);(2,8)(7,15)(9,10)(11,16)(17,20,23,36)(18,25,28,44)(19,30,33,34)(21,22,35,39)(24,40,42,43)(26,27,37,38)(29,41,45,48)(31,46,47,32);(1,4,6,5)(2,9,8,10)(3,12,13,14)(7,11,15,16)(17,21,23,35)(18,26,28,37)(19,31,33,47)(20,22,36,39)(24,41,42,48)(25,27,44,38)(29,40,45,43)(30,46,34,32)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=h,b^a=be,c^a=ch,d^a=dg,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,f^d=fi,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, 389756);