#481005

order	512
nilpotency class	4
isoclinism	isoclinic class of #480866
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×D₄	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄ ◁ C₂×#552 ◁ G
LECS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

763 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,f^2=h,b^a=be,e^a=eg,g^a=gi,c^b=ci,e^b=ei,f^b=fg,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, 388201);