#110251

order	512
nilpotency class	3
isoclinism	isoclinic class of #110242
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	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

559 conjugacy classes of subgroups; only the 144 normal subgroups are shown.

normal subgroups only

This group has 559 conjugacy classes of subgroups — too many to draw. Only its 144 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)(3,12)(4,9,17,25)(5,19)(6,10)(7,11)(8,14)(13,26,28,32)(15,23)(16,29,18,24)(20,22)(21,30,31,27)(33,34)(35,39,44,54)(36,46)(37,49)(38,41)(40,48)(42,51)(43,57,59,60)(45,56,50,53)(47,61)(52,63,64,58)(55,62);(1,3,6,14,7,15,5,20)(2,8,10,23,11,22,19,12)(4,13,18,27,17,28,16,30)(9,21,24,32,25,31,29,26)(33,35)(34,39,41,54)(36,43)(37,50)(38,44)(40,52,55,64)(42,56,49,53)(45,51)(46,60,61,57)(47,59)(48,63)(58,62);(1,4)(2,9,11,25)(3,13)(5,16)(6,18)(7,17)(8,21,22,31)(10,24,19,29)(12,26,23,32)(14,27)(15,28)(20,30)(33,36,37,48,38,47,51,62)(34,40,42,46,41,55,49,61)(35,43,50,63,44,59,45,58)(39,52,56,60,54,64,53,57)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=gi,d^b=dh,e^b=ei,f^b=fi,d^c=di,e^c=ehi,f^c=fhi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 17447);