#188147

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

2820 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,d^b=dhi,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, 95343);