#119410

order	512
nilpotency class	3
isoclinism	isoclinic class of #119409
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	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

568 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)(3,7,4,12)(5,10)(6,9)(8,13,16,11)(14,15)(17,18)(19,27,30,31)(20,21)(22,25)(23,26)(24,28,32,29);(1,3,5,4)(2,7,10,12)(6,11,14,13)(8,15,16,9)(17,19)(18,24)(20,31)(21,29)(22,28)(23,30)(25,27)(26,32);(1,4,5,3)(2,8)(6,13,14,11)(7,15)(9,12)(10,16)(17,20,21,26,23,25,22,18)(19,24,28,27,30,32,29,31)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=d,d^2=i,e^2=hi,f^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fi,g^b=ghi,e^c=ei,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26606);