#233639

order	512
nilpotency class	3
isoclinism	isoclinic class of #10078
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1249 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)(7,15)(8,17)(9,12)(10,13)(11,14)(16,19)(18,20)(21,22,26,29)(23,30,32,35)(24,28,25,33)(27,34,36,31);(1,2)(3,4)(5,7)(6,8)(9,16)(10,17)(11,18)(12,15)(13,20)(14,19)(21,23,26,32)(22,27,29,36)(24,31,25,34)(28,35,33,30);(2,4)(5,8)(6,7)(9,15)(10,20)(11,19)(12,16)(13,17)(14,18)(21,22)(23,27)(24,28)(25,33)(26,29)(30,34)(31,35)(32,36);(5,9,11,10)(6,12,14,13)(7,16,18,17)(8,15,19,20)(21,24,26,25)(22,28,29,33)(23,31,32,34)(27,35,36,30)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=h,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,c^b=cg,e^b=eh,f^b=fi,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, 140835);