#129056

order	512
nilpotency class	3
isoclinism	isoclinic class of #1812
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄²	#117
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1767
prime radical	C₂³	C₄×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄² ◁ C₂×#322 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

296 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,e^2=g,f^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 36252);