#193103

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

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

normal subgroups only

This group has 912 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) = 256

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

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

Polycyclic presentation:

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

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, 100299);