#480800

order	512
nilpotency class	4
isoclinism	isoclinic class of #480662
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	#2404	C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×#41 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

866 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)(19,27)(20,24)(21,34,35,25)(22,26)(23,30,37,44)(28,41)(29,42)(31,45)(32,47,48,39)(33,40)(36,43,38,46);(1,3)(2,7)(4,11)(5,12)(6,10)(8,15)(9,14)(13,16)(17,19)(18,23)(20,31)(21,30)(22,29)(24,38)(25,27)(26,37)(28,33)(32,46)(34,42)(35,44)(36,40)(39,45)(41,47)(43,48);(3,10)(7,14)(11,12)(15,16)(17,20,22,33)(18,24,26,40)(19,28,29,31)(21,32,35,48)(23,36,37,38)(25,39,34,47)(27,41,42,45)(30,43,44,46);(1,4,6,5)(2,8,9,13)(3,11,10,12)(7,15,14,16)(19,29)(23,37)(27,42)(28,31)(30,44)(36,38)(41,45)(43,46)

Polycyclic presentation:

2^9|a^2=f,c^2=i,d^2=h,f^2=h,b^a=be,d^a=dh,e^a=eg,g^a=gi,c^b=chi,d^b=di,f^b=fg,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, 387996);