#188126

order	512
nilpotency class	3
isoclinism	isoclinic class of #188049
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×#538	C₂
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

2826 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 95322);