#244187

order	512
nilpotency class	3
isoclinism	isoclinic class of #232783
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	C₂³	#525
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

536 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 536 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=ef,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=dhi,e^b=eh,f^b=fh,d^c=dh,e^c=ei,f^c=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, 151383);