#214605

order	512
nilpotency class	3
isoclinism	isoclinic class of #214000
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

920 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,c^2=h,d^2=hi,f^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,f^b=fh,g^b=gh,f^c=fh,g^c=ghi,g^d=gh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121801);