#188091

order	512
nilpotency class	3
isoclinism	isoclinic class of #188035
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2808 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,g^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, 95287);