#191523

order	512
nilpotency class	3
isoclinism	isoclinic class of #19713
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

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

756 conjugacy classes of subgroups.

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

Polycyclic presentation:

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

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, 98719);