#244139

order	512
nilpotency class	3
isoclinism	isoclinic class of #232760
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

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

normal subgroups only

This group has 558 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)(33,34,37,41,38,42,52,58)(35,43,46,54,45,56,50,39)(36,47,48,55,49,57,51,40)(44,53,61,64,60,63,62,59);(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)(33,35,38,45)(34,39,42,54)(36,44,49,60)(37,50,52,46)(40,53,55,63)(41,56,58,43)(47,59,57,64)(48,62,51,61);(1,4,7,6)(2,9,12,11)(3,14,16,21)(5,17,20,19)(8,22,24,13)(10,25,28,27)(15,30,18,31)(23,32,26,29)(33,36,38,49)(34,40,42,55)(35,44,45,60)(37,51,52,48)(39,53,54,63)(41,57,58,47)(43,59,56,64)(46,62,50,61);(1,5,4,17,7,20,6,19)(2,10,9,25,12,28,11,27)(3,15,14,30,16,18,21,31)(8,23,13,29,24,26,22,32)(35,45)(39,54)(43,56)(44,60)(46,50)(53,63)(59,64)(61,62)

Polycyclic presentation:

2^9|a^2=e,b^2=h,c^2=h,d^2=efh,e^2=h,f^2=i,b^a=be,c^a=cf,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,e^c=ei,f^c=fi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151335);