#331629

order	512
nilpotency class	3
isoclinism	isoclinic class of #331233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

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

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

normal subgroups only

This group has 560 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)(17,18,23,31,21,26,22,27)(19,28,30,32,20,24,29,25)(33,34)(35,39)(36,46)(37,40)(38,41)(42,43)(44,47)(45,48);(1,3,5,11)(2,6,8,9)(4,10,12,16)(7,13,14,15)(17,19,21,20)(18,24,26,28)(22,29,23,30)(25,27,32,31)(33,35,37,44,38,45,36,43)(34,39,40,47,41,48,46,42);(4,12)(7,14)(10,16)(13,15)(17,20,21,19)(18,25,26,32)(22,29,23,30)(24,27,28,31)(33,34)(35,42)(36,40)(37,46)(38,41)(39,43)(44,48)(45,47);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(19,20)(24,28)(25,32)(29,30)(33,36,38,37)(34,40,41,46)(35,43,45,44)(39,47,48,42)

Polycyclic presentation:

2^9|a^2=ef,b^2=eg,c^2=h,d^2=ehi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=di,c^b=cg,d^b=dh,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, 238825);