#236019

order	512
nilpotency class	3
isoclinism	isoclinic class of #234079
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×#532	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1324 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:

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

Polycyclic presentation:

2^9|b^2=hi,c^2=h,d^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=di,e^b=eh,f^b=fi,d^c=dh,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, 143215);