#494393

order	512
nilpotency class	4
isoclinism	isoclinic class of #33568
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×B(4,𝔽₂)
commutator	C₂×Q₈	C₂³×C₄
Frattini, cosocle	C₂²×Q₈	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂×Q₈ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×#41 ◁ G
LECS	G ▷ C₂²×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

(1,2,4,3)(5,6)(7,17)(8,14)(9,12)(10,25)(11,16)(13,20)(15,23)(18,31)(19,33)(21,32)(22,28)(24,34)(26,29)(27,36)(30,35);(1,3,4,2)(6,12)(9,23)(13,27)(14,28)(15,16)(17,33)(19,24)(22,26)(25,29)(30,31)(32,34)(35,36);(5,7,11,21)(6,13,16,31)(8,20,10,18)(9,24,23,19)(12,27,15,30)(14,32,25,17)(22,35,26,36)(28,34,29,33);(5,8,11,10)(6,14,16,25)(7,18,21,20)(9,22,23,26)(12,28,15,29)(13,17,31,32)(19,35,24,36)(27,33,30,34)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,d^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,e^b=ei,f^b=fhi,d^c=di,e^c=eh,f^c=fi,f^e=fi

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, 401589);