#133057

order	512
nilpotency class	4
isoclinism	isoclinic class of #132549
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂×C₄⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#392	D₄
LCS	G ▷ C₂×C₄⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

491 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 491 conjugacy classes of subgroups — too many to draw. Only its 74 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) = 64

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=f,c^2=g,d^2=h,e^2=i,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=egi,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fg,e^d=ei,f^d=fi,f^e=fi

Conjugacy classes & character table

41 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 40253);