#116909

order	512
nilpotency class	3
isoclinism	isoclinic class of #116906
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

542 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 542 conjugacy classes of subgroups — too many to draw. Only its 126 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)(3,11)(4,15,17,8)(5,18)(6,10)(7,13)(9,19)(12,25,27,23)(14,24)(16,22,31,30)(20,26,29,32)(21,28)(33,34,38,44,39,43,37,42)(35,40,46,54,47,55,52,59)(36,48,50,41,51,58,49,57)(45,60,61,56,62,64,63,53);(1,3,5,13,6,14,19,28)(2,7,9,11,10,21,18,24)(4,12,16,26,17,27,31,32)(8,20,22,23,15,29,30,25)(33,35)(34,40)(36,45)(37,52)(38,46)(39,47)(41,56)(42,59)(43,55)(44,54)(48,60)(49,63)(50,61)(51,62)(53,57)(58,64);(1,4)(2,8)(3,12)(5,16)(6,17)(7,20)(9,22)(10,15)(11,23)(13,26)(14,27)(18,30)(19,31)(21,29)(24,25)(28,32)(33,36,39,51)(34,41,43,57)(35,45,47,62)(37,50,38,49)(40,53,55,56)(42,58,44,48)(46,63,52,61)(54,64,59,60)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=i,d^2=h,e^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fi,c^b=cf,e^b=ei,e^c=ei,g^c=ghi

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, 24105);