#140263

order	512
nilpotency class	4
isoclinism	isoclinic class of #12465
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1677
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#472
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

213 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 84 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) = 128

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfgh,c^2=dh,d^2=gi,e^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,g^a=gi,e^b=ei,d^c=di,e^c=egh,h^c=hi

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, 47459);