#333181

order	512
nilpotency class	3
isoclinism	isoclinic class of #333127
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³	#560
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 561 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:

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

Polycyclic presentation:

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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240377);