#128554

order	512
nilpotency class	4
isoclinism	isoclinic class of #2180
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₈	#442
commutator	C₈	C₂×C₄×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂	#8170
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₈ ◁ C₂²×C₈ ◁ #1726 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

387 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=e,d^2=g,f^2=h,g^2=i,b^a=bd,d^a=dg,f^a=fg,g^a=gi,h^a=hi,c^b=cg,e^b=ei,d^c=di,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72283926, C₂×#128554
maximal normal subgroup of	#72283926, C₂×#128554
maximal quotient of	C₂×#128554

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 35750);