#147604

order	512
nilpotency class	4
isoclinism	isoclinic class of #129435
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂⁵	C₂²×C₄
Frattini, prime radical, cosocle	C₂³×D₄	C₂³
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂×#518 ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 1010 conjugacy classes of subgroups — too many to draw. Only its 138 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) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=h,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,g^b=gh,e^c=ei,g^d=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54800);