#214707

order	512
nilpotency class	3
isoclinism	isoclinic class of #214704
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	C₂³×C₄	C₂⁴
prime radical	#35692	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=g,d^2=hi,f^2=i,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=fi,g^d=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 121903);