#128863

order	512
nilpotency class	4
isoclinism	isoclinic class of #11867
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1519
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂⁴	C₂×SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

399 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 399 conjugacy classes of subgroups — too many to draw. Only its 108 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 — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=ei,d^2=gi,f^2=i,g^2=i,h^2=i,b^a=bd,d^a=dg,f^a=fghi,g^a=gi,h^a=hi,c^b=cgh,d^b=dh,g^b=gi,f^c=fi,f^d=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36059);