#236085

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
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	C₂×#3207	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1406 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)(7,16)(8,13)(9,14)(10,19)(11,15)(12,17)(18,20)(21,22)(23,30)(24,28)(25,34)(26,29)(27,33)(31,35)(32,36);(1,2)(3,4)(5,7)(6,12)(8,11)(9,17)(10,18)(13,14)(15,20)(16,19)(24,25)(28,34)(30,36)(31,33);(2,4)(5,8,11,7)(6,13,15,16)(9,17,10,18)(12,19,20,14)(21,23)(22,27)(24,33)(25,31)(26,32)(28,36)(29,35)(30,34);(5,9,11,10)(6,14,15,19)(7,17,8,18)(12,13,20,16)(21,24,26,25)(22,28,29,34)(23,31,32,33)(27,30,35,36)

Polycyclic presentation:

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

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, 143281);