#318128

order	512
nilpotency class	3
isoclinism	isoclinic class of #288899
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	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

756 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|b^2=egh,d^2=eh,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^c=dh,g^c=gi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225324);