#141069

order	512
nilpotency class	4
isoclinism	isoclinic class of #125269
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂×#493	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

367 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 367 conjugacy classes of subgroups — too many to draw. Only its 90 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)(5,6,10,15,27,16,36,35,11,17,30,19,9,14,26,32)(7,18,13,34,20,24,28,29,22,12,31,33,21,8,23,25);(5,7,10,28,27,21,36,13,11,22,30,23,9,20,26,31)(6,12,32,34,14,8,19,29,17,18,35,33,16,24,15,25);(1,2,3,4)(5,8,9,12,11,24,27,18)(6,13,16,28,17,31,14,23)(7,19,20,32,22,15,21,35)(10,29,26,34,30,25,36,33)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=eghi,d^2=ghi,e^2=i,f^2=hi,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ei,f^b=fhi,h^b=hi,d^c=di,f^c=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, 48265);