#190289

order	512
nilpotency class	3
isoclinism	isoclinic class of #190224
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

999 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)(3,6)(4,7)(5,8)(9,12)(10,13)(11,14)(15,16)(17,18,21,25,22,23,20,28)(19,24,26,32,27,30,29,31);(1,3,5,11)(2,6,8,14)(4,10,12,16)(7,13,9,15)(18,23)(20,21)(24,30)(26,29);(2,7)(4,12)(6,13)(8,9)(10,16)(14,15)(17,19)(18,24,23,30)(20,26)(21,29)(22,27)(25,31,28,32);(2,8)(6,14)(7,9)(13,15)(17,20,22,21)(18,25,23,28)(19,26,27,29)(24,31,30,32)

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=e,d^2=h,f^2=hi,g^2=h,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, 97485);