#149722

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#382	C₂³
prime radical	C₂³	C₂×#99
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2327 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

266 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 68 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,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=ch,d^b=dhi,e^b=efhi,f^b=fi,d^c=di,e^d=ehi,f^e=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 56918);