#141177

order	512
nilpotency class	4
isoclinism	isoclinic class of #125419
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1514
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

444 conjugacy classes of subgroups; only the 88 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 88 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfh,c^2=e,d^2=gh,e^2=i,f^2=gh,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fhi,g^b=gi,d^c=di,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48373);