#142071

order	512
nilpotency class	4
isoclinism	isoclinic class of #125092
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 270 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2)(3,7)(4,5)(6,8)(9,10,14,20,13,11,22,25,16,21,23,31,15,19,17,32)(12,26,28,39,27,18,34,35,30,24,37,38,29,36,33,40);(2,5)(6,7)(9,11,23,20,15,21,22,25,16,19,14,31,13,10,17,32)(12,24,37,35,29,36,34,38,30,26,28,40,27,18,33,39);(1,3,4,8)(2,6,5,7)(9,12,15,29,16,30,13,27)(10,18,11,24,21,36,19,26)(14,28,17,33,23,37,22,34)(20,35,25,38,31,40,32,39)

Polycyclic presentation:

2^9|a^2=f,b^2=defg,c^2=ei,d^2=gh,e^2=i,f^2=g,g^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,h^a=hi,d^b=dh,f^b=fh,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, 49267);