#149436

order	512
nilpotency class	4
isoclinism	isoclinic class of #129424
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂³×Q₈	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂³×Q₈ ▷ C₂² ▷ C₂ ▷ C₁

817 conjugacy classes of subgroups; only the 142 normal subgroups are shown.

normal subgroups only

This group has 817 conjugacy classes of subgroups — too many to draw. Only its 142 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=di,e^b=ei,g^b=ghi,d^c=di,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, 56632);