#214008

order	512
nilpotency class	3
isoclinism	isoclinic class of #214000
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	C₂²×#41	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

891 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=h,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,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, 121204);