#494257

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1048 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401453);