#186767

order	512
nilpotency class	3
isoclinism	isoclinic class of #18712
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

2060 conjugacy classes of subgroups; only the 338 normal subgroups are shown.

normal subgroups only

This group has 2060 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)(21,22)(23,32)(24,33)(25,29)(26,30)(27,31)(28,34)(35,36);(2,4)(5,7,9,15)(6,12,8,14)(10,17,11,16)(13,20,18,19)(21,23)(22,28)(24,31)(25,33)(26,34)(27,35)(29,32)(30,36);(5,8,9,6)(7,14,15,12)(10,13,11,18)(16,19,17,20)(21,24,27,28)(22,23,31,35)(25,32,26,36)(29,34,30,33);(21,25,27,26)(22,29,31,30)(23,33,35,34)(24,36,28,32)

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,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, 93963);