#190336

order	512
nilpotency class	3
isoclinism	isoclinic class of #190224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1013 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=e,d^2=i,f^2=hi,g^2=i,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,d^b=dh,g^b=gh,f^c=fi,g^c=gh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97532);