#233429

order	512
nilpotency class	3
isoclinism	isoclinic class of #233367
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
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

1352 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140625);