#115087

order	512
nilpotency class	3
isoclinism	isoclinic class of #115084
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1438
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

733 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 733 conjugacy classes of subgroups — too many to draw. Only its 130 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 — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,d^2=i,e^2=hi,f^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fh,c^b=cf,d^b=di,e^b=eh,f^b=fi,g^b=ghi,e^c=ehi,f^c=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, 22283);