#480821

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	#2431	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

855 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:

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

Polycyclic presentation:

2^9|a^2=f,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=ch,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, 388017);