#236065

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	C₂²×#127	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1440 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:

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

Polycyclic presentation:

2^9|b^2=h,c^2=h,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, 143261);