#480827

order	512
nilpotency class	4
isoclinism	isoclinic class of #480684
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×C₄⋊C₄	C₂⁴
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

835 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,d^2=hi,e^2=i,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,e^b=ei,f^b=fgi,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, 388023);