#318130

order	512
nilpotency class	3
isoclinism	isoclinic class of #288899
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	#2537	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 670 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)(7,15)(8,11)(9,19)(10,14)(12,17)(13,16)(18,20)(21,22)(23,26)(24,28)(25,27)(29,33)(30,34)(31,36)(32,35);(5,7,16,18,10,17,9,8)(6,11,19,12,14,20,13,15)(21,23,29,30,25,32,24,31)(22,26,33,34,27,35,28,36);(2,4)(7,17)(8,18)(11,20)(12,15)(23,30)(24,29)(26,34)(28,33)(31,32)(35,36);(5,8,9,17,10,18,16,7)(6,12,13,11,14,15,19,20)(22,27)(26,35)(28,33)(34,36)

Polycyclic presentation:

2^9|b^2=egi,d^2=e,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, 225326);