#108591

order	512
nilpotency class	3
isoclinism	isoclinic class of #8610
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

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

normal subgroups only

This group has 426 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)(3,11,14,16,12,15,13,8)(17,18,21,27,23,28,22,26)(19,29,31,24,33,40,32,41)(20,34,37,25,38,43,36,44)(30,39,47,45,35,42,46,48);(2,7,10,9)(3,12)(8,11,16,15)(13,14)(17,19,22,32,23,33,21,31)(18,24,27,40,28,41,26,29)(20,35,37,46,38,30,36,47)(25,42,34,45,44,39,43,48);(1,3,4,13,6,12,5,14)(2,8,9,11,10,16,7,15)(17,20,21,36,23,38,22,37)(18,25,26,43,28,44,27,34)(19,30,31,46,33,35,32,47)(24,39,29,45,41,42,40,48)

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,c^b=ch,f^c=fhi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15787);