#480910

order	512
nilpotency class	4
isoclinism	isoclinic class of #480884
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×D₄	C₂²⋊C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

825 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)(19,27,30,42)(20,24)(21,34,36,25)(22,26)(23,29)(28,43,31,45)(32,47,48,40)(33,41)(35,38)(37,44)(39,46);(17,19,22,30)(18,23,26,38)(20,31,33,28)(21,35,36,29)(24,39,41,37)(25,42,34,27)(32,44,48,46)(40,43,47,45);(1,3)(2,6)(4,10)(5,9)(7,13)(8,12)(11,15)(14,16)(17,20,22,33)(18,24,26,41)(19,28,30,31)(21,32,36,48)(23,37,38,39)(25,40,34,47)(27,43,42,45)(29,44,35,46);(3,9)(6,12)(10,15)(13,16)

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=i,e^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=ei,f^b=fgi,d^c=dh,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, 388106);