#233655

order	512
nilpotency class	3
isoclinism	isoclinic class of #233642
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#120
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1244 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140851);