#490398

order	512
nilpotency class	4
isoclinism	isoclinic class of #490107
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#447
commutator, Frattini, cosocle	C₄×C₈	C₂⁴
prime radical	#3169	C₂²
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

645 conjugacy classes of subgroups; only the 136 normal subgroups are shown.

normal subgroups only

This group has 645 conjugacy classes of subgroups — too many to draw. Only its 136 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) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 64 points:

(1,2,7,12)(3,13,15,24)(4,18,21,25)(5,11,6,10)(8,17,23,16)(9,19,26,20)(14,28,29,22)(27,30,32,31)(33,34,40,46)(35,47,36,42)(37,54,55,44)(38,56,58,43)(39,59,57,45)(41,52,62,49)(48,64,51,61)(50,63,53,60);(1,3)(2,8)(4,14)(5,17)(6,16)(7,15)(9,22)(10,24)(11,13)(12,23)(18,32)(19,31)(20,30)(21,29)(25,27)(26,28)(33,35,40,36)(34,41,46,62)(37,52,55,49)(38,48,58,51)(39,53,57,50)(42,56,47,43)(44,60,54,63)(45,64,59,61);(1,4,7,21)(2,9,12,26)(3,14,15,29)(5,20,6,19)(8,22,23,28)(10,18,11,25)(13,27,24,32)(16,31,17,30)(33,36,40,35)(34,42,46,47)(37,48,55,51)(38,52,58,49)(39,53,57,50)(41,56,62,43)(44,61,54,64)(45,63,59,60);(3,15)(8,23)(13,24)(14,29)(16,17)(22,28)(27,32)(30,31)(33,37,39,38,40,55,57,58)(34,43,45,44,46,56,59,54)(35,48,50,52,36,51,53,49)(41,60,61,42,62,63,64,47)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=hi,d^2=gh,e^2=gi,f^2=h,g^2=i,b^a=be,c^a=cf,d^a=dghi,e^a=eg,f^a=fh,g^a=gi,d^b=dg,e^b=eg,f^b=fh,g^b=gi,d^c=di,e^c=eh,f^c=fh

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 397594);