#127686

order	512
nilpotency class	4
isoclinism	isoclinic class of #1355
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#329
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂×#97	C₂³
socle	C₂²	C₂×#329
prime radical	C₂⁴	#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

470 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=i,d^2=g,f^2=h,h^2=i,b^a=bd,d^a=dg,f^a=fgh,h^a=hi,d^b=dh,g^b=gi,f^d=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, 34882);