#194533

order	512
nilpotency class	3
isoclinism	isoclinic class of #19050
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

1080 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)(5,6,10,14,11,15,9,13)(7,16,17,20,8,18,19,12)(21,22)(23,29,31,36)(24,26)(25,28)(27,30,34,33)(32,35);(1,2,3,4)(5,7,11,8)(6,12,15,20)(9,17,10,19)(13,16,14,18)(22,26)(25,32)(27,34)(30,33);(5,8,11,7)(6,12,15,20)(9,17,10,19)(13,18,14,16)(21,23)(22,27)(24,31)(25,33)(26,34)(28,36)(29,35)(30,32);(5,9,11,10)(6,13,15,14)(7,17,8,19)(12,16,20,18)

Polycyclic presentation:

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

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, 101729);