#190233

order	512
nilpotency class	3
isoclinism	isoclinic class of #190230
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	C₂³×C₄	C₂⁴
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1126 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 97429);