#214036

order	512
nilpotency class	3
isoclinism	isoclinic class of #214020
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	C₂³×D₄	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

939 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)(3,10,12,6)(4,9)(5,8)(7,11,16,13)(14,15)(17,18,21,26,22,25,20,24)(19,27,29,31,30,23,28,32);(1,3)(2,6)(4,13)(5,12)(7,15)(8,10)(9,16)(11,14);(1,3)(2,7)(4,13)(5,12)(6,9)(8,16)(10,15)(11,14)(17,19,22,30)(18,23,25,27)(20,29,21,28)(24,32,26,31);(1,3)(2,6)(4,11)(5,12)(7,9)(8,10)(13,14)(15,16)(17,20,22,21)(18,24,25,26)(19,28,30,29)(23,31,27,32)

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=i,f^2=h,g^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,d^c=di,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, 121232);