#479947

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂²×C₄×C₈
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

535 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,e^2=g,f^2=h,h^2=i,b^a=bi,d^a=di,c^b=ci,d^c=dg

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 387143);