#128922

order	512
nilpotency class	4
isoclinism	isoclinic class of #128808
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1762
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂²×D₈	C₂×C₄
LCS	G ▷ C₂×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₄×#97 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

400 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 108 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=ghi,g^2=i,b^a=bd,d^a=dg,f^a=fgh,g^a=gi,c^b=cg,d^b=dh,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36118);