#479934

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	C₂⁴	C₄×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

561 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:

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

Polycyclic presentation:

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

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, 387130);