#343106

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	#3549	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 641 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) = 128

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

(1,2,7,13,6,12,5,11)(3,14,18,8,17,27,16,26)(4,19,23,28,22,9,21,10)(15,30,32,29,31,25,20,24)(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);(1,3,6,17)(2,8,12,26)(4,20,22,32)(5,18,7,16)(9,25,19,30)(10,29,28,24)(11,27,13,14)(15,23,31,21)(33,35,38,47)(34,39,41,55)(36,44,51,63)(37,45,52,46)(40,53,56,60)(42,54,58,43)(48,59,57,64)(49,61,50,62);(1,4)(2,9)(3,15)(5,21)(6,22)(7,23)(8,24)(10,13)(11,28)(12,19)(14,25)(16,20)(17,31)(18,32)(26,29)(27,30)(33,36)(34,40)(35,44)(37,49)(38,51)(39,53)(41,56)(42,57)(43,59)(45,61)(46,62)(47,63)(48,58)(50,52)(54,64)(55,60);(1,4)(2,10)(3,15)(5,23)(6,22)(7,21)(8,25)(9,13)(11,19)(12,28)(14,24)(16,32)(17,31)(18,20)(26,30)(27,29)(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,b^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, 250302);