#494456

order	512
nilpotency class	4
isoclinism	isoclinic class of #33539
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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×#472	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1426 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:

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

Polycyclic presentation:

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

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, 401652);