#340042

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, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

604 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 604 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=hi,c^2=hi,d^2=ehi,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, 247238);