#339998

order	512
nilpotency class	3
isoclinism	isoclinic class of #331791
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×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

662 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 662 conjugacy classes of subgroups — too many to draw. Only its 146 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

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

Polycyclic presentation:

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

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 247194);