#236059

order	512
nilpotency class	3
isoclinism	isoclinic class of #234044
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1668 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|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, 143255);