#119208

order	512
nilpotency class	3
isoclinism	isoclinic class of #119200
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

560 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=d,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,g^b=ghi,e^c=ehi,f^c=fhi,g^c=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, 26404);