#186800

order	512
nilpotency class	3
isoclinism	isoclinic class of #18682
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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	C₂⁴	#120
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2056 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 2056 conjugacy classes of subgroups — too many to draw. Only its 338 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) = 256

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=hi,c^2=h,d^2=i,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,d^c=di,e^c=eh,f^c=fh

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, 93996);