#511886

order	512
nilpotency class	4
isoclinism	isoclinic class of #511753
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#2190	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1085 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)(7,15)(8,18)(9,21)(11,23)(13,28)(14,27)(16,31)(17,32)(19,22)(20,33)(24,34)(25,35)(26,36)(29,30);(1,2,3,4)(5,6,10,12)(7,11,16,24)(8,13,20,26)(9,14,21,27)(15,23,31,34)(17,25,22,30)(18,28,33,36)(19,29,32,35);(5,7)(6,11)(8,19)(9,22)(10,16)(12,24)(13,29)(14,30)(15,18)(17,21)(20,32)(23,28)(25,27)(26,35)(31,33)(34,36);(6,12)(8,20)(11,24)(14,27)(15,31)(18,33)(19,32)(25,30)

Polycyclic presentation:

2^9|b^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,d^b=di,f^c=fh,g^c=gi,h^c=hi,f^d=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 419082);