#133075

order	512
nilpotency class	4
isoclinism	isoclinic class of #132541
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×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 529 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=i,c^2=g,d^2=h,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,e^c=ei,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, 40271);