#235861

order	512
nilpotency class	3
isoclinism	isoclinic class of #234006
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×#121	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1359 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)(7,11)(8,16)(9,13)(10,15)(12,18)(14,20)(17,19)(21,22,26,31)(23,32,33,36)(24,30,25,29)(27,35,28,34);(1,2)(3,4)(5,7)(6,10)(8,17)(9,14)(11,16)(12,20)(13,19)(15,18)(21,23,26,33)(22,27,31,28)(24,35,25,34)(29,36,30,32);(2,4)(5,7)(6,11)(8,15)(9,14)(10,16)(12,19)(13,20)(17,18)(21,23)(22,28)(24,35)(25,34)(26,33)(27,31)(29,32)(30,36);(7,14)(10,19)(11,20)(15,17)(23,33)(27,28)(32,36)(34,35)

Polycyclic presentation:

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

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, 143057);