#107753

order	512
nilpotency class	3
isoclinism	isoclinic class of #9415
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#386
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

716 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 716 conjugacy classes of subgroups — too many to draw. Only its 186 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)(17,18,21,27,23,28,22,26)(19,29,31,24,33,39,32,40)(20,34,35,42,36,41,37,25)(30,38,44,43,46,48,45,47);(1,3,6,2)(4,11,13,8)(5,9,7,10)(12,15,14,16)(17,19,22,32,23,33,21,31)(18,24,27,39,28,40,26,29)(20,30,35,44,36,46,37,45)(25,38,41,47,42,48,34,43);(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)(23,36)(24,38)(26,34)(27,41)(28,42)(29,43)(31,45)(32,44)(33,46)(39,47)(40,48)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 14949);