#331654

order	512
nilpotency class	3
isoclinism	isoclinic class of #331304
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

576 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 576 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 512

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=egh,c^2=i,d^2=eh,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=di,d^c=de,f^c=fh,g^c=gi

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, 238850);