#194517

order	512
nilpotency class	3
isoclinism	isoclinic class of #19014
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

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

1140 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 101713);