#235937

order	512
nilpotency class	3
isoclinism	isoclinic class of #234028
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

1309 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)(7,12)(8,13)(9,11)(10,14)(15,18)(16,20)(17,19)(21,22,26,23)(24,31,32,36)(25,30,35,28)(27,34,29,33);(1,2)(3,4)(6,11)(8,9)(12,18)(13,14)(15,17)(19,20)(21,23,26,22)(24,29,32,27)(25,28,35,30)(31,34,36,33);(2,4)(5,7)(6,12)(8,17)(9,15)(10,16)(11,18)(13,19)(14,20)(21,24)(22,27)(23,29)(25,34)(26,32)(28,36)(30,31)(33,35);(5,8,10,9)(6,13,14,11)(7,15,16,17)(12,18,20,19)(24,32)(27,29)(31,36)(33,34)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^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=dh,e^b=eh,f^b=fi,d^c=dhi,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, 143133);