#194603

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1294 conjugacy classes of subgroups; only the 258 normal subgroups are shown.

normal subgroups only

This group has 1294 conjugacy classes of subgroups — too many to draw. Only its 258 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)(5,6,10,14,11,7,9,13)(8,16,19,12,17,20,18,15)(21,22)(23,30,32,35)(24,33)(25,28)(26,29)(27,31,36,34);(1,2,3,4)(5,7,11,6)(8,15,17,12)(9,13,10,14)(16,19,20,18)(21,22)(23,27)(24,28)(25,33)(26,29)(30,34)(31,35)(32,36);(5,8,11,17)(6,12,7,15)(9,18,10,19)(13,16,14,20)(21,23)(22,27)(24,31)(25,34)(26,32)(28,35)(29,36)(30,33);(8,17)(12,15)(16,20)(18,19)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)

Polycyclic presentation:

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

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, 101799);