#124411

order	512
nilpotency class	4
isoclinism	isoclinic class of #11989
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1551
commutator	C₄²	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#100
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂²×C₄ ▷ C₂ ▷ C₁

287 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 287 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=fh,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cf,d^a=df,f^a=fi,g^a=gi,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 31607);