#352092

order	512
nilpotency class	3
isoclinism	isoclinic class of #352081
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2856	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

764 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 764 conjugacy classes of subgroups — too many to draw. Only its 182 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)(6,10)(8,15)(11,16)(12,17)(14,19)(18,20)(23,29)(24,30)(26,32)(28,33)(31,35)(34,36);(5,6,8,10,9,12,15,17)(7,11,14,16,13,18,19,20)(21,22,25,27)(23,26,31,34)(24,28,30,33)(29,32,35,36);(5,7,9,13)(6,11,12,18)(8,14,15,19)(10,16,17,20)(21,23,30,35,25,31,24,29)(22,26,33,36,27,34,28,32);(2,4)(6,12)(7,13)(10,17)(14,19)(22,27)(26,34)(28,33)(32,36)

Polycyclic presentation:

2^9|b^2=ei,c^2=fhi,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,d^b=dhi,d^c=dh

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 259288);