#246974

order	512
nilpotency class	3
isoclinism	isoclinic class of #246957
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#52084	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

997 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 997 conjugacy classes of subgroups — too many to draw. Only its 146 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=hi,d^2=h,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,d^b=di,e^b=eh,f^b=fhi,e^c=ei,f^c=fi,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 154170);