#149713

order	512
nilpotency class	5
isoclinism	isoclinic class of #13605
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2127
commutator	C₄²	C₂²×C₈
Frattini, cosocle	#382	C₂³
prime radical	C₂×#127	C₂×C₄
LCS	G ▷ C₄² ▷ C₂×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂×C₄² ◁ #2328 ◁ G
LECS	G ▷ #382 ▷ C₂²×C₄ ▷ C₂² ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 329 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=g,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=dh,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, 56909);