#480999

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
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	C₂×#553	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#553 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

794 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=gi,d^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,f^b=fgi,d^c=dhi,f^e=fi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 388195);