#188130

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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=hi,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, 95326);