#188041

order	512
nilpotency class	3
isoclinism	isoclinic class of #188035
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#379
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁶ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂² ▷ C₁

2992 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)(3,11,13,7)(4,15,16,8)(5,9)(6,10)(12,25,27,19)(14,24,29,20)(17,31,32,21)(18,22)(23,26,30,28);(1,3,6,14)(2,7,10,20)(4,12,16,28)(5,13,18,29)(8,19,15,30)(9,11,22,24)(17,26,32,27)(21,23,31,25);(1,4,5,17)(2,8,9,21)(3,12,13,26)(6,16,18,32)(7,19,11,23)(10,15,22,31)(14,28,29,27)(20,30,24,25);(3,13)(7,11)(12,26)(14,29)(19,23)(20,24)(25,30)(27,28)

Polycyclic presentation:

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

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, 95237);