#214915

order	512
nilpotency class	3
isoclinism	isoclinic class of #214768
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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	#42794	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

877 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 32 points:

(1,2)(3,7)(4,9)(5,14)(6,10)(8,11)(12,16)(13,15)(17,18,22,25,20,26,21,29)(19,24,28,23,27,31,30,32);(1,3)(2,7)(4,13)(5,11)(6,12)(8,14)(9,15)(10,16)(19,27)(21,22)(24,31)(25,29);(1,3)(2,8)(4,11)(5,13)(6,12)(7,14)(9,16)(10,15)(17,19)(18,23)(20,27)(21,30)(22,28)(24,29)(25,31)(26,32);(1,4,6,5)(2,9,10,14)(3,11,12,13)(7,8,16,15)(17,19)(18,24)(20,27)(21,28)(22,30)(23,29)(25,32)(26,31)

Polycyclic presentation:

2^9|a^2=g,d^2=hi,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, 122111);