#205930

order	512
nilpotency class	3
isoclinism	isoclinic class of #205761
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#390
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

935 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,c^2=fi,e^2=hi,f^2=hi,g^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fhi,e^b=ei,g^b=gi,d^c=dh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 113126);