#233855

order	512
nilpotency class	3
isoclinism	isoclinic class of #233836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#392
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1115 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fh,g^b=ghi,e^c=ei,f^c=fi,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 141051);