#482835

order	512
nilpotency class	4
isoclinism	isoclinic class of #482734
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²×#99
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄⋊C₄	C₂⁴
prime radical	C₂³	C₂²×C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈ ◁ C₂×#557 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

724 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=hi,d^2=hi,e^2=h,f^2=i,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=eh,f^b=fgh,d^c=dhi,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, 390031);