#282688

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#550	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

789 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 189884);