#146751

order	512
nilpotency class	4
isoclinism	isoclinic class of #12741
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

946 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(5,6)(7,18,21,37)(8,23)(9,27)(10,32)(11,17)(12,20,38,57)(13,24)(14,30)(15,45)(16,35)(19,52,54,36)(22,43,33,46)(25,47)(26,40)(28,49,58,39)(29,41)(31,48)(34,44)(42,62)(50,61)(51,67,60,53)(55,65,59,66)(56,63,68,64);(1,2,3,4)(5,7)(6,12)(8,19)(9,28)(10,33)(11,21)(13,36)(14,43)(15,46)(16,49)(17,38)(18,48)(20,45)(22,35)(23,60)(24,56)(25,59)(26,54)(27,37)(29,53)(30,57)(31,58)(32,39)(34,55)(40,51)(41,66)(42,52)(44,63)(47,64)(50,65)(61,67)(62,68);(5,8,11,26)(6,13,17,42)(7,19,21,54)(9,29,31,61)(10,34,35,25)(12,36,38,52)(14,44,15,47)(16,50,32,41)(18,51,37,60)(20,56,57,68)(22,59,33,55)(23,48,40,27)(24,30,62,45)(28,53,58,67)(39,66,49,65)(43,63,46,64)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=i,d^2=h,e^2=h,h^2=i,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,h^a=hi,d^b=dhi,e^b=ehi,f^b=fi,h^b=hi,d^c=dhi,e^c=ehi,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 53947);