#133025

order	512
nilpotency class	4
isoclinism	isoclinic class of #132573
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

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

normal subgroups only

This group has 492 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:

(1,2,6,13,32,45,24,11)(3,16,18,48,52,64,22,58)(4,21,8,14,33,9,7,15)(5,25,28,44,60,43,23,41)(10,36,37,55,63,62,40,19)(12,31,35,26,61,27,42,29)(17,49,30,47,20,50,54,46)(34,56,39,59,57,51,38,53);(2,9)(3,17)(4,6)(5,26)(10,37)(11,15)(12,42)(13,14)(16,46)(18,22)(19,55)(20,52)(21,45)(24,33)(25,41)(29,60)(30,54)(34,38)(35,61)(36,53)(39,57)(40,63)(43,44)(47,64)(48,50)(49,58)(51,56)(59,62);(1,3)(2,10,14,38)(4,22)(5,19)(6,30)(7,20)(8,17)(9,34,13,37)(11,40,21,57)(12,16,25,49)(15,39,45,63)(18,33)(23,59)(24,54)(26,56)(27,62)(28,53)(29,51)(31,36)(32,52)(35,48,44,47)(41,46,42,58)(43,50,61,64)(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=eg,g^a=gi,c^b=ci,d^b=dh,e^b=egh,f^b=fgh,d^c=dghi,f^c=fgi,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, 40221);