#214739

order	512
nilpotency class	3
isoclinism	isoclinic class of #214736
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

1166 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)(3,14,18,8)(4,19,21,10)(5,9)(6,11)(7,13)(12,16)(15,28,30,24)(17,25,22,32)(20,27)(23,26)(29,31);(1,3)(2,8)(4,16)(5,22)(6,18)(7,15)(9,25)(10,12)(11,14)(13,24)(17,23)(19,29)(20,30)(21,31)(26,32)(27,28);(2,9)(3,15)(5,16)(8,19)(10,14)(11,26)(12,13)(18,30)(23,31)(24,32)(25,28)(27,29);(1,4,20,17)(2,10,27,32)(3,16,30,23)(5,18,31,15)(6,21,7,22)(8,12,28,26)(9,14,29,24)(11,19,13,25)

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=cei,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, 121935);