#511882

order	512
nilpotency class	4
isoclinism	isoclinic class of #511751
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×#575	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1106 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1106 conjugacy classes of subgroups — too many to draw. Only its 184 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=dg,f^a=fh,h^a=hi,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, 419078);