#244171

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	C₂²×D₄	#41
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 587 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:

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

Polycyclic presentation:

2^9|a^2=e,b^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, 151367);