#122210

order	512
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	?
homology	Abelianization (H₁) = C₂×C₈² · Schur multiplier (H₂) = C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₈²	C₂³
commutator, socle	C₂²	C₂×C₈²
prime radical	C₂³	C₈²
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₈² ◁ G
LECS	G ▷ C₈² ▷ C₄² ▷ C₂² ▷ C₁

217 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 64
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 64 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,d^2=f,e^2=g,f^2=h,g^2=i,c^a=chi,c^b=ch

Conjugacy classes & character table

224 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 29406);