#214933

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
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	C₂×C₄²	C₂⁴
prime radical	#42880	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

892 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)(3,6)(4,7)(5,9)(8,12)(10,14)(11,13)(15,16)(17,18,22,28,23,27,20,25)(19,29,30,38,32,24,36,40)(21,34,33,42,35,26,37,41)(31,39,46,47,45,43,44,48);(1,3)(2,6)(4,11)(5,10)(7,13)(8,15)(9,14)(12,16)(17,19,23,32)(18,24,27,29)(20,30,22,36)(21,31,35,45)(25,40,28,38)(26,39,34,43)(33,44,37,46)(41,48,42,47);(2,7)(4,12)(6,13)(8,9)(11,16)(14,15)(17,20,23,22)(18,25,27,28)(19,30,32,36)(21,33,35,37)(24,38,29,40)(26,41,34,42)(31,44,45,46)(39,47,43,48);(3,10)(6,14)(11,16)(13,15)(17,21,23,35)(18,26,27,34)(19,31,32,45)(20,33,22,37)(24,39,29,43)(25,41,28,42)(30,44,36,46)(38,47,40,48)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,c^b=ce,d^b=di,f^b=fh,g^b=gh,f^c=fi,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, 122129);