#110102

order	512
nilpotency class	3
isoclinism	isoclinic class of #110086
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂³×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator, prime radical	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 557 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) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,f^2=hi,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=di,e^b=eh,f^b=fh,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17298);