#480590

order	512
nilpotency class	4
isoclinism	isoclinic class of #480454
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	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂×C₂²⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

858 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387786);