#190266

order	512
nilpotency class	3
isoclinism	isoclinic class of #190208
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

1030 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)(3,10)(4,8)(5,13)(6,9)(7,11)(12,15)(14,16)(17,18,21,26,22,23,20,25)(19,24,27,31,28,30,29,32);(18,23)(20,21)(24,30)(27,29);(1,3)(2,7)(4,11)(5,14)(6,12)(8,15)(9,16)(10,13)(17,19)(18,24,23,30)(20,27)(21,29)(22,28)(25,31,26,32);(1,4,6,5)(2,8,9,13)(3,11,12,14)(7,15,16,10)(17,20,22,21)(18,25,23,26)(19,27,28,29)(24,31,30,32)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97462);