#317234

order	512
nilpotency class	3
isoclinism	isoclinic class of #317219
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#535
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 472 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 — 4 generators on 64 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 224430);