#107768

order	512
nilpotency class	3
isoclinism	isoclinic class of #8528
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#379
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	C₂²⋊#95
prime radical	C₂⁴	#123
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

730 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=i,e^2=h,f^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,d^b=di,e^b=ehi,f^b=fhi,d^c=dhi,e^c=eh,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, 14964);