#126580

order	512
nilpotency class	3
isoclinism	isoclinic class of #447
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂×D₄
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	C₂×#336
prime radical	C₂⁴	#97
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

445 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=i,e^2=g,f^2=h,h^2=i,b^a=bd,d^a=di,f^a=fi,c^b=cg

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 33776);