#108590

order	512
nilpotency class	3
isoclinism	isoclinic class of #8608
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#380
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1434
prime radical	C₂⁴	C₄×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

446 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,10,4,7)(3,11,13,16,14,15,12,8)(17,18,21,27,23,28,22,26)(19,29,31,24,33,39,32,40)(20,34,36,25,37,41,35,42)(30,38,45,43,46,48,44,47);(2,7,10,9)(8,15,16,11)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,36,45,37,46,35,44)(25,38,34,47,42,48,41,43);(1,3,4,12,6,14,5,13)(2,8,9,11,10,16,7,15)(17,20,21,35,23,37,22,36)(18,25,26,41,28,42,27,34)(19,30,31,44,33,46,32,45)(24,38,29,43,40,48,39,47)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=d,d^2=h,e^2=h,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,f^c=fhi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15786);