#381619

order	512
nilpotency class	3
isoclinism	isoclinic class of #29902
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	#2622
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂	#14303
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=f,d^2=h,f^2=i,h^2=i,b^a=be,c^a=cf,f^a=fi,c^b=cg,e^b=ei,g^b=gi,d^c=de,f^c=fi,g^d=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, 288815);