#149989

order	512
nilpotency class	5
isoclinism	isoclinic class of #149927
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	#382	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

297 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 297 conjugacy classes of subgroups — too many to draw. Only its 54 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=gi,d^2=h,e^2=g,f^2=i,b^a=bd,d^a=df,f^a=fh,h^a=hi,c^b=cgh,d^b=dh,e^b=efhi,f^b=fi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 57185);