#478199

order	512
nilpotency class	3
isoclinism	isoclinic class of #476144
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2675
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

946 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 946 conjugacy classes of subgroups — too many to draw. Only its 186 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 — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=f,c^2=ef,d^2=f,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,d^b=def,g^b=gi,d^c=de,e^c=ei,f^c=fi,h^c=hi,h^d=hi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 385395);