#343107

order	512
nilpotency class	3
isoclinism	isoclinic class of #343099
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2679
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#18696	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

637 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 637 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=efi,c^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=def,c^b=cg,d^b=de,e^b=eh,g^b=ghi,f^c=fi,g^c=ghi,e^d=eh,f^d=fi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 250303);