#190385

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

1057 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=e,d^2=hi,f^2=hi,g^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, 97581);