#244173

order	512
nilpotency class	3
isoclinism	isoclinic class of #232789
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#532	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 585 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 — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=fi

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, 151369);