#118874

order	512
nilpotency class	3
isoclinism	isoclinic class of #118874
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1440
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	#3444	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

519 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=d,d^2=i,e^2=h,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,f^b=fh,g^b=ghi,f^c=fhi,g^c=ghi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 26070);