#244188

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
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	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

594 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 594 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151384);