#141131

order	512
nilpotency class	4
isoclinism	isoclinic class of #125300
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1505
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂²×M₁₆	C₂³
prime radical	#33179	C₂
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×M₁₆ ▷ C₂×C₄ ▷ C₂ ▷ C₁

515 conjugacy classes of subgroups; only the 96 normal subgroups are shown.

normal subgroups only

This group has 515 conjugacy classes of subgroups — too many to draw. Only its 96 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)(5,6,11,16,10,17,9,15,12,18,27,28,13,7,19,21)(8,20,24,34,23,14,22,30,25,31,35,29,26,32,33,36);(5,7,13,6,12,17,10,18)(8,20,26,14,25,31,23,32)(9,15,27,16,19,21,11,28)(22,29,35,30,33,34,24,36);(1,2,3,4)(5,8,13,26,12,25,10,23)(6,14,17,31,18,32,7,20)(9,22,11,24,19,33,27,35)(15,29,28,36,21,34,16,30)

Polycyclic presentation:

2^9|a^2=f,b^2=dfhi,c^2=ei,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,d^b=dh,f^b=fhi,g^b=gi,f^d=fi

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, 48327);