#191256

order	512
nilpotency class	3
isoclinism	isoclinic class of #8790
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#382
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	#125
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

518 conjugacy classes of subgroups; only the 254 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 254 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=fh,c^2=e,d^2=hi,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^b=ghi,g^c=ghi

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, 98452);