#317334

order	512
nilpotency class	3
isoclinism	isoclinic class of #317315
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	#553
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

439 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 439 conjugacy classes of subgroups — too many to draw. Only its 118 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=i,d^2=ei,e^2=h,f^2=i,g^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,f^b=fh,d^c=dh,f^c=fi,g^c=gi

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, 224530);